Transcription

1 maxima Contents maxima 1.. maxima 1.3. maxima maxima maxima Gram-Schmidt

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3 maxima maxima. maxima maxima MIT( ) Project MAC Macsyma Macsyma MAC s SYmbolic MAnipulator MIT Lisp Project MAC Macsyma Symbolics Inc. Macsyma DOE Macsyma Symbolics Macsyma Symbolics Inc. Macsyma Inc 1999 Texas William Schelter Doe Macsyma GNU Common Lisp 1990 maxima MIT Bill Schelter 001 UCB R.Fateman GPL maxima Asir Maple Mathematica MuPAD Reduce Yacas Maple Mathematica Reduce Macsyma Asir Yacas GPL maxima maxima Asir MuPAD Yacas 1.. maxima. maxima Linux FreeBSD MacOS X Unix OS Common Lisp Windows OS maxima maxima Windows cygwin Unix Unix maxima Windows maxima maxima URL Windows maxima

4 4 maxima 1.3. maxima. Unix Terminal maxima i i i i i i i ooooo o ooooooo ooooo ooooo I I I I I I I o 8 8 I \ + / I \ -+- / ooooo 8oooo o 8 8 o ooooo 8oooooo ooo8ooo ooooo 8 Copyright (c) Bruno Haible, Michael Stoll 199, 1993 Copyright (c) Bruno Haible, Marcus Daniels Copyright (c) Bruno Haible, Pierpaolo Bernardi, Sam Steingold 1998 Copyright (c) Bruno Haible, Sam Steingold Maxima Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. This is a development version of Maxima. The function bug_report() provides bug reporting information. (C1) Unix xmaxima Windows maxima (C1) (C1) ( ) maxima Unix maxima TeXmacs Emacs+imaxima TeXmacs maxima (C1) integrate(1/(1+x^3),x); ş ť (D1) log (x x+1) + arctan x log(x+1) 3 (C) a : (1 + x - 3*y)^5; (D) ( 3y + x + 1) 5 (C3) expand(d); (D3) 43y xy y 4 70x y 3 540xy 3 70y x 3 y + 70x y + 70xy + 90y 15x 4 y 60x 3 y 90x y 60xy 15y + x 5 + 5x x x + 5x + 1 (C4) 10!; (D4) (C5) factor(d4); (D5)

5 maxima 5 (C*) (D*) maxima maxima (C3) expand(d); d (D) (C*) quit(); maxima (C) B:matrix([1,:}); Incorrect syntax: Missing ] B:matrix([1,:}) ^ (C) Incorrect syntax: Premature termination of input at ;. ; ^ (C) B:matrix([1,]); (D) [ 1 ] (C3) A+B; FULLMAP found arguments with incompatible structure. -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) *** - POSITION: :START = 1 should not be greater than :END = 0 The following restarts are available: R1 = Macsyma top-level 1. Break [1]> maxima 1. Break [1]> R1 1. Break [1]> (run) maxima maxima 1. Break [1]> (bye)

6 maxima. maxima (maxima ) factor describe( factor ) (C1) describe("factor"); 0: DONTFACTOR :(maxima.info)definitions for Matrices and Linear Algebra. 1: EXPANDWRT_FACTORED :Definitions for Simplification. : FACTOR :Definitions for Polynomials. 3: FACTORFACSUM :Definitions for Simplification. 4: FACTORFLAG :Definitions for Polynomials. 5: FACTORIAL :Definitions for Number Theory. 6: FACTOROUT :Definitions for Polynomials. 7: FACTORSUM :Definitions for Polynomials. 8: GCFACTOR :Definitions for Polynomials. 9: GFACTOR :Definitions for Polynomials. 10: GFACTORSUM :Definitions for Polynomials. 11: MINFACTORIAL :Definitions for Number Theory. 1: NEXTLAYERFACTOR :Definitions for Simplification. 13: NUMFACTOR :Definitions for Special Functions. 14: SAVEFACTORS :Definitions for Polynomials. 15: SCALEFACTORS :Definitions for Miscellaneous Options. 16: SOLVEFACTORS :Definitions for Equations. Enter n, all, none, or multiple choices eg 1 3 : ( ) Enter n, all, none, or multiple choices eg 1 3 : 5 Info from file /usr/local/info/maxima.info: - Function: FACTORIAL (X) The factorial function. FACTORIAL(X) = X!. See also MINFACTORIAL and FACTCOMB. The factorial operator is!, and the double factorial operator is!!. (D1) FALSE factor maxima tutorial manual info xmaxima Windows maxima, PDF URL Maxima Reference Manual DOE-Maxima Refernce The Maxima Book

7 maxima 7 Maxima Reference Manual (maxima-5.6)

8 8. maxima 1 a 11 a 1 a 1n a A = 1 a a n a m1 a m a mn maxima A : matrix([a 11,, a 1n ],[a 1,, a n ],, [a m1,, a mn ]); 1. (C1) ( A : ) matrix([a, b], [c, d]); a b (D1) c d. (C) ( B : matrix([-1, ), 3], [0, 4, -1]); 1 3 (D) maxima : a ij = a[i, j] maxima A : genmatrix(a, i, j, i1, j1); a i1,j1 a i1,j A =..... a i,j1 a i,j i1=j1= 1 A:genmatrix(a, i, j); 3. (C1) a[i,j] := i + j - 1; (D1) a i,j := i + j 1 (C) A : genmatrix(a, 3, 4); (D) a[i,j] := i,j. 7 3 maxima entermatrix 4. (C1) A : entermatrix(,3); Row 1 Column 1: 1; Row 1 Column : 0;

9 Row 1 Column 3: -1; Row Column 1: 4; Row Column : ; Row Column 3: 8; Matrix( entered. ) (D1) 4 8 maxima 9 5. (C) B : entermatrix(3,3); Is the matrix 1. Diagonal. Symmetric 3. Antisymmetric 4. General Answer 1,, 3 or 4 : 1; Row 1 Column 1: -1; Row Column : ; Row 3 Column 3: 4; Matrix entered. (D) B (C1) ( A : matrix([1, ) ], [3, 4]); 1 (D1) 3 4 (C) ( B : A; ) 1 (D) 3 4 (C3) ( B; ) 1 (D3) 3 4 diagmatrix 6. (C1) A : diagmatrix(4, ); (D1) n ident ident(n); 7. (C1) (D1) ident(3);

10 10 (m, n)- zeromatrix A : zeromatrix(m, n); 8. (C1) C : zeromatrix(3, ); (D1) (i, j)- a (m, n) ematrix(m, n, a, i, j) 9. (C1) D : ematrix(4, 4, -1,, 3); (D1) (m, 1) m m columnvector columnvector load(eigen); 10. (C1) load(eigen); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) /usr/local/share/maxima/5.9.0/share/matrix/eigen.mac (C) a : columnvector([1,, 3]); (D) 1 3 (C3) x : [1,, 3, 4]; (D3) [1,, 3, 4] (C4) b : columnvector(x); 1 (D4) 3 4

11 maxima A, B A + B A + B A, B a 11 a 1 a 1n b 11 b 1 b 1n a 1 a a n b 1 b b n A =, B = a ij b ij a m1 a m a mn b m1 b m b mn a 11 + b 11 a 1 + b 1 a 1n + b 1n a 1 + b 1 a + b a n + b n A + B = a ij + b ij a m1 + b m1 a m + b m a mn + b mn maxima A B C = A + B C : A + B; 11. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) ( B : matrix([1, ) ], [3, 4]); 1 (D) 3 4 (C3) ( C : A + B; ) a + 1 b + (D3) c + 3 d + 4 A c ca c*a; 1. (C4) ( 3*A; ) 3a 3b (D4) 3c 3d (C5) ( f*a; ) af bf (D5) cf df (C6) ( n*b; ) n n (D6) 3n 4n

12 A = (a i ) (m, l) B = (b ij ) (l, n) A B AB = (c ij ) AB (m, n) c ij (1) () l c ij = a ik b kj k=1 = a i1 b 1j + a i b j + + a il b lj AB BA AB = BA A,B n AB,BA n AB = BA maxima. A B AB maxima A.B AB 13. (C1) ( A : matrix([, ) 3, 4], [-1, 0, ]); 3 4 (D1) 1 0 (C) B : matrix([3, 0], [-1, ], [3, 4]); (D) (C3) ( C : A.B; ) 15 (D3) 3 8 (C4) D : B.A; (D4) (C5) C.D; incompatible dimensions - cannot multiply -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) (C6) ( E : matrix([a, ) b], [c, d]); a b (D6) c d (C7) ( F : matrix([e, ) f], [g, h]); e f (D7) g h (C8) ( E.F; ) bg + ae bh + af (D8) dg + ce dh + cf (C9) F.E;

13 (D9) ( cf + ae df + be ) ch + ag dh + bg maxima A n A k A k := AA k 1 maxima A A k Aˆˆk k Aˆ k k A k 14. (C1) ( A : ) matrix([, 3], [0, -1]); 3 (D1) 0 1 (C) ( A^^5; ) 3 33 (D) 0 1 (C3) B : matrix([a, b, c], [d, e, f], [g, h, i]); (D3) a b c d e f g h i (C4) B^3; (D4) a3 b 3 c 3 d 3 e 3 f 3 g 3 h 3 i 3 B 3 (C6) B^^3; (D6) [ ] [ g (c i + b f + a c) + d (c h + b e + a b) + a (c g + b d + a ) ] [ ] Col 1 = [ ] [ g (f i + e f + c d) + d (f h + e + b d) + a (f g + d e + a d) ] [ ] [ ] [ g (i + f h + c g) + d (h i + e h + b g) + a (g i + d h + a g) ] [ ] [ h (c i + b f + a c) + e (c h + b e + a b) + b (c g + b d + a ) ] [ ] Col = [ ] [ h (f i + e f + c d) + e (f h + e + b d) + b (f g + d e + a d) ] [ ] [ ] [ h (i + f h + c g) + e (h i + e h + b g) + b (g i + d h + a g) ]

14 14 [ ] [ i (c i + b f + a c) + f (c h + b e + a b) + c (c g + b d + a ) ] [ ] Col 3 = [ ] [ i (f i + e f + c d) + f (f h + e + b d) + c (f g + d e + a d) ] [ ] [ ] [ i (i + f h + c g) + f (h i + e h + b g) + c (g i + d h + a g) ]

15 maxima A AX = XA = I A X A A 1 I ( E ) maxima A A 1 maxima Aˆˆ(-1); invert(a); 15. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) ( A^^(-1); ) d b (D) ad bc ad bc c a ad bc ad bc (C3) ( B : matrix([1, ) ], [, 1]); 1 (D3) 1 (C4) ( B^^(-1); ) 1 (D4) 3 3 (C5) (D5) ( invert(b); ) = ad bc = 0 A 1 maxima Cramer Cramer Cramer maxima expand(adjoint(a))/expand(determinant(a)) A Cramer adjoint(a) A determinant(a) A expand 16. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) ( expand(adjoint(a))/expand(determinant(a)); ) d b (D) ad bc ad bc c a ad bc ad bc (C3) C : matrix([a, b, c], [d, e, f], [g, h, i]); (D3) a b c d e f g h i (C4) ratsimp( invert(c) - expand(adjoint(c))/expand(determinant(c)) );

16 16 (D6) ratsimp

17 maxima n x 1, x,..., x n m a 11 x 1 + a 1 x + + a 1n x n = b 1 a 1 x 1 + a x + + a n x n = b (3) a m1 x 1 + a m x + + a mn x n = b m maxima (3) solve solve ( ) (C1) solve([,,..., m], [,,..., n]); eq1 : solve([eq1, eq,..., eq], [x, y,..., z]); 17. (C1) eq1 : a*x + b*y = 0; (D1) by + ax = 0 (C) eq : c*x + d*y = 0; (D) dy + cx = 0 (C3) solve([eq1, eq], [x, y]); (D3) [[x = 0, y = 0]] (C4) eq3 : a*x + b*y = e; (D4) by + ax = e (C5) eq4 : c*x + d*y = f; (D5) dy + cx = f (C6) solve([eq3, eq4], [x, y]); (D6) [[ x = de bf, y = ]] ce af bc ad bc ad (C7) solve([3*x-*y+z=3, x+y-3*z=7, y+z=9], [x, y, z]); (D7) [[ x = 6, y = 36, z = ]] (C8) solve([3*x-*y+z=3, x+y-3*z=7], [x, y]); (D8) [[ ]] x = 5z+17, y = 10z (C9) solve([3*x - *y = 3, x + y = 7, x - y = 9], [x, y]); Inconsistent equations: (3) -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) (C8) z (C9) maxima 5... (3)

18 18 a 11 a 1 a 1n x 1 b 1 a A = 1 a a n......, x = x., b = b. a m1 a m a mn x n b m (3) (4) Ax = b (3) A (3) a 11 a 1 a 1n b 1 a (A b) := 1 a a n b a m1 a m a mn b m (3) maxima x, y,..., z eq1, eq,..., eqm (C1) A : coefmatrix([eq1, eq,..., eqm], [x, y,..., z]); A (C1) Ab : augcoefmatrix([eq1, eq,..., eqm], [x, y,..., z]); Ab 3. maxima augcoefmatrix (A b) (A b) solve a 11 a 1 a 1n a A = 1 a a n......, a m1 a m a mn (1) () (3) 3 A A A =

19 maxima 19 A A A A A rank(a) (3) rank(a) = rank(a b) maxima echelon echelon(a); rank rank(a); 18. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) rank(a); (D) (C3) ( echelon(a); ) 1 b (D3) a 0 1 (C4) B : matrix([1,, 3], [4, 5, 6], [7, 8, 9]); (D4) (C5) rank(b); (D5) (C6) echelon(b); (D6) (C7) C : matrix([1,, 3], [4, 5, 6], [7, 8, 0]); (D7) (C8) rank(c); (D8) 3 (C9) echelon(c); (D9) (C9) 19. (C1) eq1 : 3*x - *y = 3; (D1) 3x y = 3 (C) eq : x + y = 7; (D) y + x = 7 (C3) eq3 : x - y = 9;

20 0 (D3) x y = 9 (C4) A : coefmatrix([eq1, eq, eq3], [x, y]); (D4) (C5) Ab : augcoefmatrix([eq1, eq, eq3], [x, y]); 3 3 (D5) (C6) rank(a); (D6) (C7) rank(ab); (D7) 3 (C8) echelon(a); (D8) (C9) (D9) echelon(ab); solve

21 maxima (4) b = 0 Ax = 0 rank(a) = rank(a 0) (4) n rank(a) n = rank(a) x 1 = x = = x n = 0 (4) m = n A n n = rank(a) det(a) 0 A det(a) det(a) 0 (4) x = A 1 b 0. (C1) A : matrix([1,, 3], [4, 5, 6], [7, 8, 9]); (D1) (C) rank(a); (D) (C3) solve([x+*y+3*z=0, 4*x+5*y+6*z=0, 7*x+8*y+9*z=0], [x,y,z]); Dependent equations eliminated: (3) (D3) [[x = %R1, y = %R1, z = %R1]] (C4) eq1 : x + *y + 3*z = 0; (D4) 3z + y + x = 0 (C5) eq : 4*x + 5*y + 6*z = 0; (D5) 6z + 5y + 4x = 0 (C6) eq3 : 6*x + 7*y = 0; (D6) 7y + 6x = 0 (C7) B : coefmatrix([eq1, eq, eq3], [x, y, z]); (D7) (C8) rank(b); (D8) 3 (C9) solve([eq1, eq, eq3], [x, y, z]); (D9) [[x = 0, y = 0, z = 0]] (C10) eq4 : x + *y + 3*z = 10; (D10) 3z + y + x = 10 (C11) eq5 : 4*x + 5*y + 6*z = 11; (D11) 6z + 5y + 4x = 11 (C1) eq6 : 7*x + 8*y = 1; (D1) 8y + 7x = 1 (C13) C : coefmatrix([eq4, eq5, eq6], [x, y, z]);

22 (D13) (C14) Cb : augcoefmatrix([eq4, eq5, eq6], [x, y, z]); (D14) (C15) rank(c) - rank(cb); (D15) 0 (C16) bb : matrix([10], [11], [1]); (D16) (C17) xx : C^^(-1). bb; (D17) (C18) solve([eq4, eq5, eq6], [x, y, z]); (D18) [[ x = 8, y = 9, z = 0]] 3 3 (C19) determinant(a); (D19) 0 (C0) determinant(b); (D0) 4 (C1) determinant(c); (D1) 7 4. x = A 1 b solve

23 maxima A n A (n, n) a 11 a 1 a 1n a A = 1 a a n a n1 a n a nn A det(a) det(a) := sgn(σ)a 1σ(1) a σ() a nσ(n) σ S n S n n maxima determinant (1),(3) maxima triangularize maxima triangularize () 1. (C1) A : matrix([1,, 3], [4, 5, 6], [7, 8, 9]); (D1) (C) B : triangularize(a); (D) (C3) determinant(a); (D3) 0 (C4) determinant(b); (D4) 0 (C5) C : matrix([1,, 3], [4, 5, 6], [7, 8, 0]); (D5) (C6) D : triangularize(c); (D6) (C7) determinant(c); (D7) 7 (C8) determinant(d); (D8) 567 (C9) F : matrix([a, b, c], [d, e, f], [g, h, i]);

24 4 (D9) a b c d e f g h i (C10) determinant(f); (D10) a (ei fh) b (di fg) + c (dh eg) (C11) G : triangularize(f); (D11) a b c 0 ae bd af cd 0 0 (ae bd) i + (cd af) h + (bf ce) g (C1) ratsimp( determinant(f) - col(row(g, 3), 3) ); (D1) ( 0 ) col row col( row(m, i), j) M (i, j) maxima A mattrace mattrace(a); load( nchrpl ). (C1) load("nchrpl"); (D1) /usr/local/share/maxima/5.9.0/share/matrix/nchrpl.mac (C) ( A : matrix([a, ) b], [c, d]); a b (D) c d (C3) mattrace(a); (D3) d + a (C4) B : matrix([a, b, c], [d, e, f], [g, h, i]); (D4) a b c d e f g h i (C5) mattrace(b); (D5) i + e + a (C6) sum(col(row(b, i), i), i, 1, 3); (D6) ( i + e + a ) Caylay-Hamilton maxima ( ) a b A = c d A tr(a)a + det(a)i = O 3. (C7) ( I :) diagmatrix(, 1); 1 0 (D7) 0 1 (C8) A^^ - mattrace(a)*a + determinant(a)*i;

25 (D8) (C9) (D9) maxima 5 ( ) a (d + a) + ad + a b (d + a) + bd + ab c (d + a) + cd + ac d d (d + a) + ad ( ratsimp(d8); ) A det(a) tr(a) G A (det(g) 0) det(g 1 AG) = det(a), tr(g 1 AG) = tr(a) A B det(ab) = det(a) det(b) tr(a + B) = tr(a) + tr(b) tr(ab) = tr(ba) maxima 4. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) ( G : matrix([s, ) t], [u, v]); s t (D) u v (C3) ( determinant(a) -) ( determinant(g^^(-1) ) (. A. G); (D3) s(du+cs) u(bu+as) v(bv+at) t(dv+ct) (bu+as)v sv tu sv tu sv tu sv tu sv tu ad bc (C4) ratsimp(d3); (D4) 0 (C5) B : matrix([a, b, c], [d, e, f], [g, h, i]); (D5) a b c d e f g h i (C6) (D6) F : matrix([o, p, q], [r, s, t], [u, v, w]); o p q r s t u v w t(du+cs) sv tu ) ( s(dv+ct) sv tu (C7) ratsimp( determinant(b) - determinant(f^^(-1). B. F) ); (D7) 0 (C8) load("nchrpl"); (D8) /usr/local/share/maxima/5.9.0/share/matrix/nchrpl.mac (C9) ratsimp(mattrace(a) - mattrace(g^^(-1). A. G)); (D9) 0 (C10) ratsimp(mattrace(b) - mattrace(f^^(-1). B. F)); (D10) 0 ) u(bv+at) + sv tu

26 6 5. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) ( B : matrix([s, ) t], [u, v]); s t (D) u v (C3) ratsimp(determinant(a. B) - determinant(a) * determinant(b)); (D3) 0 (C4) C : matrix([a,b,c],[d,e,f],[g,h,i]); (D4) a b c d e f g h i (C5) D : matrix([o,p,q],[r,s,t],[u,v,w]); (D5) o p q r s t u v w (C6) ratsimp(determinant(c. D) - determinant(c) * determinant(d)); (D6) 0 6. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) ( B : matrix([s, ) t], [u, v]); s t (D) u v (C3) load("nchrpl"); (D3) /usr/local/share/maxima/5.9.0/share/matrix/nchrpl.mac (C4) mattrace(a. B); (D4) dv + bu + ct + as (C5) mattrace(b. A); (D5) dv + bu + ct + as (C6) mattrace(a. B) - mattrace(b. A); (D6) 0 (C7) C : matrix([a, b, c], [d, e, f], [g, h, i]); (D7) a b c d e f g h i (C8) D : matrix([o, p, q], [r, s, t], [u, v, w]); (D9) o p q r s t u v w (C10) mattrace(c. D); (D10) iw + fv + cu + ht + es + br + gq + dp + ao (C11) mattrace(d. C); (D11) iw + fv + cu + ht + es + br + gq + dp + ao (C1) mattrace(c. D) - mattrace(d. C);

27 (D1) 0 maxima 7 1. A newdet newdet determinant determinant newdet 6... A A t A maxima A transpose 7. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) ( At : transpose(a); ) a c (D) b d (C3) B : matrix([a, b, c], [d, e, f], [g, h, i]); (D3) a b c d e f g h i (C4) Bt : transpose(b); (D4) a d g b e h c f i (C5) ( C : matrix([1, ), 3, 4], [5, 6, 7, 8]); (D5) (C6) Ct : transpose(c); 1 5 (D6) A A = t A A A A = t A A A B = A + t A, C = A t A B C A A = B + C maxima maxima 8. (C1) symm(x) := (1/)*(X + transpose(x)); (D1) symm (X) := 1 (X + TRANSPOSE (X)) (C) ( A : matrix([1, ) ], [3, 4]); 1 (D) 3 4

28 8 (C3) (D3) ( symm(a); ) (C4) B : matrix([3, -1, ], [4, -, 1], [9, 9, 5]); (D4) (C5) symm(b); (D5) (C6) altm(x) := (1/)*(X - transpose(x)); (D6) altm (X) := 1 (X TRANSPOSE (X)) (C7) ( altm(a); ) 0 1 (D7) 1 (C8) 0 altm(b); (D8) (C9) ( A - (symm(a) ) + altm(a)); 0 0 (D9) 0 0 (C10) B - (symm(b) + altm(b)); (D10) (C1) symm (C6) altm A A A maxima conjugate conjugate load(eigen); 9. (C1) load(eigen); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) /usr/local/share/maxima/5.9.0/share/matrix/eigen.mac (C) ( A : matrix([1) - %I, + 3*%I], [%I, 4]); 1 i 3i + (D) i 4 (C3) ( conjugate(a); ) i + 1 3i (D3) i 4

29 maxima 9 A A = t A A A A = t A A A A = Re + Im Re A Re + Im = A = t A = t Re t Im Re Im A Re + Im = A = t A = t Re + t Im Re Im A B = Re + t Re C = Re t Re + Im t Im, + Im + t Im, B C A = B + C B C 30. (C1) load("nchrpl"); (D1) /usr/local/share/maxima/5.9.0/share/matrix/nchrpl.mac (C) load(eigen); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D) /usr/local/share/maxima/5.9.0/share/matrix/eigen.mac (C3) symm(x) := (1/)*(X + transpose(x)); (D3) symm (X) := 1 (X + TRANSPOSE (X)) (C4) altm(x) := (1/)*(X - transpose(x)); (D4) altm (X) := 1 (X TRANSPOSE (X)) (C5) ream(x) := (1/)*(X + conjugate(x)); (D5) ream (X) := 1 (X + CONJUGATE (X)) (C6) imgm(x) := (1/)*(X - conjugate(x)); (D6) imgm (X) := 1 (X CONJUGATE (X)) (C7) herm(x) := symm(ream(x)) + altm(imgm(x)); (D7) herm (X) := symm (ream (X)) + altm (imgm (X)) (C8) skew(x) := altm(ream(x)) + symm(imgm(x)); (D8) skew (X) := altm (ream (X)) + symm (imgm (X)) (C9) ( A : matrix([1 ) + %I, - 3*%I], [, 4*%I]); i + 1 3i (D9) 4i (C10) ( herm(a); ) 1 3i (D10) 3i + 0 (C11) ( skew(a); ) i 3i (D11) 3i 4i (C1) A - (herm(a) + skew(a));

30 30 (D1) ( ) 5. maxima (maxima Lisp ) (,,...) :=... ; block (,,...) := block([ ],...,...,... N);... N

31 maxima maxima maxima maxima [a, b,..., z] [ ] maxima 31. (C1) a : [1,, 3]; (D1) [1,, 3] (C) a; (D) [1,, 3] (array) a[i, j,..., n] i, j,..., k 5 n n 3. (C3) b[i] := i; (D3) b i := i (C4) b; (D4) b (C5) b[5]; (D5) 5 (C6) c[i,j] := i*j; (D6) c i,j := ij (C7) c; (D7) c (C8) c[3,4]; (D8) 1 maxima 31 a 3 a[i] 33. (C9) a[1]; (D9) 1 (C10) a[]; (D10) (C11) a[3]; (D11) 3 (C1) listp(a); (D13) true (C14) listp(b); (D14) false

32 3 listp listp(a); a true false 34. (C1) a : [ [1,, 3], [4, 5, 6], [7, 8, 9] ]; (D1) [[1,, 3], [4, 5, 6], [7, 8, 9]] (C) a[1]; (D) [1,, 3] (C3) a[]; (D3) [4, 5, 6] (C4) a[3]; (D4) [7, 8, 9] (C5) a[1][1]; (D5) 1 (C6) a[1][]; (D6) (C7) a[1][3]; (D7) 3 (C8) a[][1]; (D8) 4 (C9) a[][]; (D9) 5 (C10) a[][3]; (D10) 6 (C11) a[3][1]; (D11) 7 (C1) a[3][]; (D1) 8 (C13) a[3][3]; (D13) 9 (C14) a[,3]; Wrong number of indices: [, 3] -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) a[i][j] (a[i])[j] 35. (C1) A : matrix([a, b, c, d], [e, f, g, h], [i, j, k, l]);

33 (D1) a b c d e f g h i j k l (C) A[1]; (D) [a, b, c, d] maxima 33 (C3) A[]; (D3) [e, f, g, h] (C4) A[3]; (D4) [i, j, k, l] (C5) A[][3]; (D5) g (C6) A[,3]; (D6) g (C7) matrixp(a); (D7) true (C8) listp(a); (D8) false (C9) matrixp(a[1]); (D9) false (C10) listp(a[1]); (D10) true (C11) B : [[a,b,c,d],[e,f,g,h],[i,j,k,l]]; (D11) [[a, b, c, d], [e, f, g, h], [i, j, k, l]] (C1) B[1]; (D1) [a, b, c, d] (C13) B[]; (D13) [e, f, g, h] (C14) B[3]; (D14) [i, j, k, l] (C15) B[][3]; (D15) g (C16) B[,3]; Wrong number of indices: [, 3] -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) (C17) matrixp(b); (D17) false (C18) listp(b); (D18) true (C19) matrixp(b[1]); (D19) false (C0) listp(b[1]); (D0) true

34 34 matrixp listp(a); a true false maxima 3 n P(i,j,n), Q(a,i,n), R(a,i,j,n), (i j) (n, k) M (1) i j P(i,j,n) M (n, k) M () i a Q(a,i,n) M (n, k) M (3) i j a R(a,i,j,n) M em.mac /* em.mac */ P(i, j, n) := block([i, A], I : ident(n), A : zeromatrix(n, n), A[i][i] : -1, A[j][j] : -1, A[i][j] : 1, A[j][i] : 1, I + A )$ Q(a, i, n) := block([a], A : ident(n), A[i][i] : a, A )$ R(a, i, j, n) := block([i,a], I : ident(n), A : zeromatrix(n,n), A[i][j] : a, I + A )$ 36. (C1) load("./em.mac")$ (C) M : matrix([1,,3],[4,5,6],[7,8,9]); (D) (C3) M1 : R(-4,, 1, 3). M; (D3) (C4) M : R(-7, 3, 1, 3). M1; (D4)

35 maxima 35 (C5) M3 : R(-, 3,, 3). M; (D5) (C6) M4 : Q((-1/3),, 3). M3; (D6) (C7) echelon(m); (D7) (C8) P(,3,5); (D8) (C9) P(,4,5); (D10) (C11) Q(3,,5); (D11) (C1) R(3,,4,5); (D1) (C13) determinant(p(,4,5)); (D13) 1 (C14) determinant(q(3,,5)); (D14) 3 (C15) determinant(r(3,,4,5)); (D15) maxima maxima row col A row(a, i); A i col(a, j);

36 36 A j row col (m, 1) (1, n) 37. (C1) (D1) A : matrix([a, b, c], [d, e, f], [g, h, i]); a b c d e f g h i (C) col(a, 1); (D) a d g (C3) matrixp(col(a, 1)); (D3) true (C4) listp(col(a, 1)); (D4) false (C5) row(a, 1); (D5) ( a b c ) (C6) matrixp(row(a, 1)); (D6) true (C7) listp(row(a, 1)); (D7) false (C8) col(a,1)[1]; (D8) [a] (C9) col(a,1)[1][1]; (D9) a (C10) col(a,1)[][1]; (D10) d (C11) col(a,1)[3][1]; (D11) g (C1) row(a,1)[1]; (D1) [a, b, c] (C13) row(a,1)[1][]; (D13) b (C14) row(a,1)[1][3]; (D14) c A[i]; A i 1 row(a, i); A[i]; transpose 1 (m, 1) transpose(a[i]); 38. (C15) A[1]; (D15) [a, b, c] (C16) matrixp(a[1]); (D16) false (C17) listp(a[1]); (D17) true

37 (C18) transpose(a[1]); maxima 37 (D18) a b c (C19) matrixp(transpose(a[1])); (D19) true col A j transpose(transpose(a)[j]); 39. (C0) (D0) a d g transpose(transpose(a)[1]); maxima matblock matreplace matblock(a, i1, i, j1, j); a 1,1 a 1,j1 a 1,j a 1,n.... a i1,1 a i1,j1 a i1,j a i1,n A =.... a i,1 a i,j1 a i,j a i,n.... a m,1 a m,j1 a m,j a m,n a i1,j1 a i1,j.. a i,1 a i,j1 matreplace(a, B, i1, i, j1, j) A b 1,j j1+1 a 1,j j1+1 B =.. a i i1+1,1 a i i1+1,j j1+1 matblock matreplace matblock.mac /* matblock.mac */ matblock(mat,i1,i,j1,j) := block([_a, _m, _n, _i, _j],

38 38 )$ _m : i - i1, _n : j - j1, _A : zeromatrix(_m + 1, _n + 1), for _i : 1 thru _m + 1 do for _j : 1 thru _n + 1 do _A[_i][_j] : Mat[i1 + _i - 1][j1 + _j - 1], _A matreplace(mat1,mat,i1,i,j1,j) := block([_a, _m, _n, _i, _j], _m : i - i1, _n : j - j1, _A : copymatrix(mat1), for _i : 1 thru _m + 1 do for _j : 1 thru _n + 1 do _A[i1 + _i - 1][j1 + _j - 1] : Mat[_i][_j], _A )$ load( matblock.mac ); 40. (C1) A : matrix([q,w,e,r,t],[y,u,i,o,p],[a,s,d,f,g],[h,i,j,k,l],[z,x,c,v,b]); q w e r t y u i o p (D1) a s d f g h i j k l z x c v b (C) load("./matblock.mac"); (D)./matblock.mac (C3) matblock(a,,4,,4); (D3) u i o s d f i j k (C4) matblock(a,,,,); (D4) ( u ) (C5) matblock(a,,5,3,4); i o (D5) d f j k c v (C6) B : matrix([1,,3],[4,5,6],[7,8,9]); (D6) (C7) ( C : matrix([1,],[3,4]); ) 1 (D7) 3 4

39 maxima 39 (C8) (D8) (C9) (D9) matreplace(a,b,,4,,4); q w e r t y 1 3 p a g h l z x c v b matreplace(a,c,1,,1,); 1 e r t 3 4 i o p a s d f g h i j k l z x c v b matreplace matreplace P(i,j,n), Q(a,i,n), R(a,i,j,n) (k,n) M (1) i 1j P(i,j,n) M () i a Q(a,i,n) M (3) i j a R(a,j,i,n) M 41. (C1) load("./em.mac"); (D1)./em.mac (C) M : matrix([1,,3],[4,5,6],[7,8,9]); (D) (C3) M1 : M. R(-, 1,, 3); (D3) (C4) M : M1. R(-3, 1, 3, 3); (D4) (C5) M3 : M. R(-,, 3, 3); (D5) (C6) M4 : M3. Q((-1/3),, 3); (D6) (C7) M. P(1, 3, 3);

40 40 (D7) addrow addcol addrow addrow(m, L1, L,...); M (m, n) L1, L,... n (l, n) M addrow addcol addcol(m, L1, L,...); M L1, L,... m (m, l) transpose(l1), transpose(l),... M addcol 4. (C1) (D1) A : matrix([a,b,c],[d,e,f],[g,h,i],[j,k,l]); a b c d e f g h i j k l (C) ( B : matrix([1,,3],[4,5,6]); ) 1 3 (D) (C3) C : matrix([1,],[3,4],[5,6],[7,8]); 1 (D3) (C4) addrow(a, C); Incompatible structure - ADDROW//ADDCOL -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) (C6) addrow(a, B); a b c d e f g h i (D6) j k l (C7) (D7) addcol(a, C); a b c 1 d e f 3 4 g h i 5 6 j k l 7 8 (C8) addrow(a, [1,, 3], [4, 5, 6]); a b c d e f g h i (D8) j k l

41 maxima 41 (C9) addcol(a, [1, 3, 5, 7], [, 4, 6, 8]); a b c 1 (D9) d e f 3 4 g h i 5 6 j k l 7 8 matblock matreplace addrow addcol A ( ) R S A = O T A,R T O det(a) = det(r) det(s) 43. (C1) ( R :) matrix([a, b], [c, d]); a b (D1) c d (C) T : matrix([r, s, t], [u, v, w], [x, y, z]); (D) r s t u v w x y z (C3) ( S : matrix([h, ) i, j], [k, l, m]); h i j (D3) k l m (C4) O : zeromatrix(3, ); (D4) (C5) A : addrow( addcol(r, S), addcol(o, T) ); a b h i j c d k l m (D5) 0 0 r s t 0 0 u v w 0 0 x y z (C6) determinant(r); (D6) ad bc (C7) determinant(t); (D7) r (vz wy) s (uz wx) + t (uy vx) (C8) ratsimp(determinant(r)*determinant(t)); (D8) ((ad bc) rv + (bc ad) su) z+((bc ad) rw + (ad bc) tu) y+((ad bc) sw + (bc ad) tv) x (C9) determinant(a); (D9) ad (r (vz wy) s (uz wx) + t (uy vx)) bc (r (vz wy) s (uz wx) + t (uy vx))

42 4 (C10) ratsimp(d8 - D9); (D10) 0 A (i, j) a ij A i,j maxima minor minor(a,i,j); B = (b ij ), b ij := ( 1) i+j det(a ji ), i, j A A adjoint adjoint(a); 44. (C1) (D1) (C) (D) (C3) (D3) A : matrix([a,b,c,d],[e,f,g,h],[i,j,k,l],[m,n,o,p]); a b c d e f g h i j k l m n o p minor(a,1,1); f g h j k l n o p minor(a,,3); a b d i j l m n p 45. (C1) ( A :) matrix([a, b], [c, d]); a b (D1) c d (C) ( B : zeromatrix(, ) ); 0 0 (D) 0 0 (C3) B[1][1] : (-1)^(1+1)*determinant(minor(A,1,1)); (D3) d (C4) B[1][] : (-1)^(1+)*determinant(minor(A,,1)); (D4) b (C5) B[][1] : (-1)^(+1)*determinant(minor(A,1,)); (D5) c (C6) B[][] : (-1)^(+)*determinant(minor(A,,)); (D6) a (C7) ( B; ) d b (D7) c a (C8) ( adjoint(a); ) d b (D8) c a (C9) ( B - adjoint(a); ) 0 0 (D9) 0 0

43 maxima 43 A 1 = A A maxima adjoint invert Lisp maxima 46. /* the following routines compute inverses and adjoints of matrices */ /* if ratmx is false [the default] then the elements will not be converted to cre-form */ adjoint(mat):= block([adj,n], n:length(mat), adj:ident(n), if n#1 then for i thru n do for j thru n do adj[i,j]:(-1)^(i+j)*determinant(minor(mat,j,i)), adj)$ invert(mat):= block([adj,ans], adj:adjoint(mat), ans:block([scalarmatrixp:true], adj/(row(mat,1).col(adj,1))), if scalarmatrixp=true and length(mat)=1 /* row(mat,1).col(adj,1) = determinant(mat) */ then ans[1,1] else ans)$ submatrix M m1, m,... n1, n,... submatrix(m1, m,..., M, n1, n,...); 47. (C1) A : matrix([a,b,c,d,e],[f,g,h,i,j],[k,l,m,n,p],[p,q,r,s,t],[u,v,w,x,y]); a b c d e f g h i j (D1) k l m n p p q r s t u v w x y (C) B : submatrix(,4,a,,3); (D) a d e k n p u x y A (i, j) a ij A[i][j]; A[i,j]; row col col(row(a, i), j); a ij (1, 1) (a ij ) col(row(a, i), j)[1][1]; col(row(a, i), j)[1,1]; 48. (C1) A : matrix([a, b, c, d], [e, f, g, h], [i, j, k, l]);

44 44 (D1) a b c d e f g h i j k l (C) A[][3]; (D) g (C3) A[,3]; (D3) g (C4) col(row(a, ), 3); (D4) ( g ) (C5) col(row(a, ), 3)[1][1]; (D5) g (C6) col(row(a, ), 3)[1,1]; (D6) g A (i, j) a ij A[i][j] : a; A[i,j] : a; A A A matreplace D : matrix([a]); matreplace(a, D, i, i, j, j); 49. (C1) (D1) A : matrix([a, b, c, d], [e, f, g, h], [i, j, k, l]); a b c d e f g h i j k l (C) A[][3] : -3; (D) 3 (C3) A; (D3) a b c d e f 3 h i j k l (C4) B : A; (D4) a b c d e f 3 h i j k l (C5) B[][3] : x; (D5) x (C6) B; (D6) (C7) A; (D7) (C8) a b c d e f x h i j k l a b c d e f x h i j k l C : copymatrix(a);

45 (D8) a b c d e f x h i j k l (C9) C[][3] : g; (D9) g (C10) C; (D10) (C11) A; a b c d e f g h i j k l maxima 45 (D11) a b c d e f x h i j k l (C1) load("./matblock.mac"); (D1)./matblock.mac (C13) D : matrix([g]); (D13) ( g ) (C14) matreplace(a, D,,, 3, 3); (D14) a b c d e f g h i j k l (C15) A; (D15) a b c d e f x h i j k l 6. (C4) (D7) B : A; B A A B A A B B A copymatrix setelmx M setelmx(a, i, j, M); M (i, j) a M M 50. (C1) (D1) (C) (D) M : matrix([a,b,c,d],[e,f,g,h],[i,j,k,l]); a b c d e f g h i j k l setelmx(x,, 3, M); a b c d e f x h i j k l

46 n K K n K n K n 1 K n K m x K n y K m x 1 x =., y =. x n y m y x y 1 = a 11 x 1 + a 1 x + + a 1n x n y = a 1 x 1 + a x + + a n x n y m = a m1 x 1 + a m x + + a mn x n x, y f y = f(x) A Mat(m, n; K)(K m n ) a 11 a 1 a 1n a A = 1 a a n a m1 a m a mn y = f(x) = Ax A f A Mat(m, n; K) K n K m f A f K n K m X Y X Y f X K n Y K m f X Y K n K m f f K n V K V + (1) v + w = w + v, v, w V () (u + v) + w = u + (v + w), u, v, w V y 1

47 maxima 47 (3) 1 0: v + 0 = 0 + v = v, v V (4) v V, 1 v V : v + ( v) = ( v) + v = 0 (5) α (β v) = (αβ) v, α, β K, v V (6) (α + β) v = α v + β v, α, β K, v V (7) α (v + w) = α v + α w, α K, v, w V (8) 1 v = v, v V (9) 0 v = 0, v V (3) 0 (4) v v ( 1)v = v V (n ) A = (a 1, a,..., a n ) v V v 1 v 1 v A =. Kn : v = v 1 a 1 + v a + + v n a n v n v 1 v = (a 1,..., a n ). = Av A v n V n A V v A a A n n K n n E = (e 1, e,..., e n ) v K n v 1 v v =. = v 1e 1 + v e + + v n e n = Ev v n K n v = v E ( ) ( )

48 48 V K n W K m f V W f(αv + βw) = αf(v) + βf(w), α, β K, v, w V V W V W V A = (a 1,..., a n ) W B = (b 1,..., b m ) f (f(a 1 ),..., f(a n )) W n v V A = (a 1,..., a n ) v = v 1 a 1 + v n a n = Av A f f(v) = v 1 f(a 1 ) + + v n f(a n ) f(a) = (f(a 1 ),..., f(a n )) f(v) = f(a)v A f(a i ) W B = (b 1,..., b m ) f(a 1 ) = a 11 a 1 + a 1 a + a m1 b m f(a ) = a 1 a 1 + a a + a m b m f(a n ) = a 1n a 1 + a n a + a mn b m A Mat(m, n; K) a 11 a 1 a 1n a A = 1 a a n a m1 a m a mn A f (A, B) f(a) = BA f(v) = BAv A f(v) B w B (f(v) = Bw B ) w B = Av A V K n W K m (A, B) f A A f (A, B) f

49 maxima V W W V ( ) v, w W αv + βw W, α, β K ( : αv + βw V ) V {0} V V S S V S S V S = {v v = α 1 v α k v k, α 1,..., α k K, v 1,..., v k S, k N} S V S S S = S V A = (a 1,..., a n ) V = {a 1,..., a n } n dim V = n dim K V = n V W dim W dim V V = W V W dim V = n dim W = m f : V W V W V Ker f := {v V f(v) = 0} f (Kernel) W Im f := {f(v) v V } f (Image) Im f Im f f V W v, w Ker f f(αv + βw) = αf(v) + βf(w) = 0 αv + βw Ker f αf(v) + βf(w) = f(αv + βw) Im f K n K m A = (a 1,..., a n ) V B = (b 1,..., b m ) W A Mat(m, n; K) f (A, B) v = Av A, v A K n v Ker f Av A = 0 v A v A Ax = 0 Ker f ( ) 5

50 V = K 3, W = K f : V W f 1 ( 0 =, f 3) 0 ( ) 1 0 =, f 0 ( 0 = 1 ) f A ( ) 0 A = 3 1 Ker f = {v K 3 Av = 0} maxima Ker f (C1) ( A : matrix([, ) 0, ],[3, -1, -]); 0 (D1) 3 1 (C) v : matrix([s],[t],[u]); (D) s t u (C3) ( o : ) matrix([0],[0]); 0 (D3) 0 (C4) eq1 : (A. v)[1][1] = o[1][1]; (D4) u + s = 0 (C5) eq : (A. v)[][1] = o[][1]; (D5) u t + 3s = 0 (C6) solve([eq1, eq],[s, t, u]); (D6) [[s = %R1, t = 5%R1, u = %R1]] Ker f = r 5r r r K = dim Ker f = 1 Im f Im f = {f(a 1 ),..., f(a n )} f(a i ) B w B,i Im f K m {w B,1,..., w B,n } a i = Aa A,i, a A,i K n, i {1,..., n} a A,i = e i, i {1,..., n}

51 maxima 51 w B,i = Aa A,i = Ae i, i {1,..., n} f i = Ae i A i A = (f 1,..., f n ) = (w B,1,..., w B,n ) Im f {f 1,..., f n } Im f A rank A A dim Im f = rank A f rank f rank f := dim Im f = rank A. 5. V 3 A = (a 1, a, a 3 ) W B = (b 1, b ) f V W f(a 1 ) = 3b 1 + b f(a ) = b 1 + b f(a 3 ) = 6b 1 4b f (A, B) A ( ) 3 6 A = 1 4 f (C1) ( A : matrix([3, ) -, 6], [, 1, -4]); 3 6 (D1) 1 4 (C) eq1 : 3*s - *t + 6*v = 0; (D) 6v t + 3s = 0 (C3) eq : *s + t - 4*v = 0; (D3) 4v + t + s = 0 (C4) solve([eq1, eq], [s, t, v]); (D4) [[ s = %R1, t = 4%R1, v = %R1 ]] 7 7 (C5) rank(a); (D5) { r Ker f = 7 a 1 + 4r } 7 a + ra 3 r K = { 7 a 1 + 4r } 7 a + a 3

52 5 dim Ker f = 1 dim Im f = rank f = rank A = Im f = W w W Im f w Im f v V w = f(v) w B = Av A Ax = w B w Im f w f(a 1 ),..., f(a n ) w B f 1,..., f n Ax = w B A (A, w B ) w Im f rank A = rank(a, w B ). 53. V = K 3 W = K 3 f : V W f(v) = Av, v V a = 1, b = 1 W 4 3 Im f (C1) A : matrix([1,, 3], [4, 5, 6], [7, 8, 9]); (D1) (C) a : transpose(matrix([-, 1, 4])); (D) 1 4 (C3) b : transpose(matrix([-, 1, 3])); (D3) 1 3 (C4) rank(a); (D4) (C5) rank(addcol(a,a)); (D5) (C6) rank(addcol(a,b)); (D6) 3 (C7) addcol(a,a);

53 maxima 53 (D7) (C8) addcol(a,b); (D8) rank A = rank(a, a) a Im f rank A rank(a, b) b / Im f f : V W Ker f dim V = dim Ker f + dim Im f. dim Ker f = dim V rank f = dim V rank A Ker f Ax = V = K 3, W = K 3 f : V W A = f(v) = Av, v V (C1) A : matrix([1,, 3], [4, 5, 6], [7, 8, 9]); (D1) (C) v : transpose(matrix([s, t, u])); (D) s t u (C3) AV : A. v; 3u + t + s (D3) 6u + 5t + 4s 9u + 8t + 7s (C4) eq1 : AV[1][1]; (D4) 3u + t + s (C5) eq : AV[][1]; (D5) 6u + 5t + 4s (C6) eq3 : AV[3][1]; (D6) 9u + 8t + 7s (C7) solve([eq1, eq, eq3], [s, t, u]); Dependent equations eliminated: (3) (D7) [[s = %R1, t = %R1, u = %R1]]

54 54 (C8) rank(a); (D8) (C9) A; (D9) (C10) load("./em.mac"); (D10)./em.mac (C11) A1 : A. R(-, 1,, 3); (D11) (C1) A : A1. R(-3, 1, 3, 3); (D1) (C13) A3 : A. R(-,, 3, 3); (D13) (C14) A4 : A3. Q((-1/3),, 3); (D14) Ker f = r r r R = 1 r 1 dim Ker f = 1 dim Im f = rank f = rank A = Im f = 1 4, 5, 3 6 = 1 4, = 1 + dim V = dim Ker f + dim Im f V n W m f V W V A = (a 1,..., a n ) A = (a 1,..., a n)

55 maxima 55 A A a i V a 1,..., a n a 1 = p 11 a 1 + p 1 a + + p n1 a n a = p 1 a 1 + p a + + p n a n a n = p 1n a 1 + p n a + + p nn a n P Mat(n, n; K) p 11 p 1 p 1n p P := 1 p p n p n1 p n p nn A = A P P A A ι: V V (A, A ) rank ι = rank P = n P V v Av A = A v A (= v) A P v A = A v A v A = P v A A A V = K n A A n n P P = A 1 A 55. V = R 3 V A = (a 1, a, a ) a 1 =, a = 3 0, a 1 = V A = (a 1, a, a ) a 1 = 1 1, a = 3 1, a 1 =

56 56 A = , A = A A P P = (A ) 1 A (C1) A1 : matrix([, 3, 1], [-, 0, 1], [1, -1, ]); (D1) (C) A : matrix([1, 3, 1], [1, 1, -], [0, 3, 4]); (D) (C3) P : A^^(-1). A1; (D3) (C4) A1 - (A. P); (D4) P = v = A A v v A v A v A = A 1 v, (C5) v : transpose(matrix([-1, 3, 6])); (D5) (C6) v1 : A1^^(-1). v; (D6) (C7) v : A^^(-1). v; v A = A 1 v

57 (D7) (C8) P. v1; (D8) (C9) v -P. v1; maxima 57 (D9) v A = , v A = v A = P v A W B B B B Q Mat(n, n; K) B = B Q Q f : V W (A, B) (A, B ) A A f(a) = BA, f(a ) = B A f A = Q 1 A P, B QA = BA = f(a) = f(a P ) = f(a )P = B A P QA = A P A = QAP 1 rank A = rank A

58 W = K W B = (b 1, b ) B = (b 1, b ) ( ) ( ( ( ) b 1 =, b =, b 1) 1 =, b 1) = B B ( ) ( ) B =, B 1 = 1 B B Q Q = B 1 B f : V W (A, B) ( ) 0 1 F = f (A, B ) F (C1) A1 : matrix([, 3, 1], [-, 0, 1], [1, -1, ]); (D1) (C) A :matrix([1, 3, 1], [1, 1, -], [0, 3, 4]); (D) (C3) P : A^^(-1). A1; (D3) (C4) ( B1 : matrix([-1, ) 3], [, 1]); 1 3 (D4) 1 (C5) ( B : matrix([1, ) ], [1, -]); 1 (D5) 1 (C6) ( Q : B^^(-1). B1; 1 ) (D6) (C7) ( F1 : matrix([, ) 0, -1], [4, 1, -5]); 0 1 (D7) (C8) ( F : Q. F1. (P^^(-1)); ) 3 (D8) (C9) ( Q^^(-1). F ). P; 0 1 (D9) 4 1 5

59 maxima 59 ( 1 ) Q = 3 4 ( F = QF P 1 3 = )

60 a b a b a b θ a b a b := a b cos θ a = (a 1, a, a 3 ) b = (b 1, b, b 3 ) a b = a 1 b 1 + a b + a 3 c 3 V V V K g a 1, a, b 1, b V α 1, α, β 1, β K g(α 1 a 1 +α a, β 1 b 1 +β b ) = α 1 β 1 g(a 1, b 1 )+α 1 β g(a 1, b )+α β 1 g(a, b 1 )+α β g(a, b ) a, b V g(a, b) = g(b, a) g(a, a) 0 a = 0 maxima K C R V n V A = {a 1, a,, a n } a V a = α 1 a 1 + α a + + α n a n α 1 α a A =. K α n

61 maxima 61 a A V a b V A α 1 α a A =., b β A =. α n β n g(a, b) g(a, b) = α 1 β 1 g(a 1, a 1 ) + α 1 β g(a 1, a ) + + α n β n g(a n, a n ) n n = α i β j g(a i, a j ) i=1 j=1 β 1 g ij := g(a i, a j ), i, j {1,,..., n} g i,j g 11 g 1 g 1n g G := 1 g g n g n1 g n g nn g A G G g A g G g 11 g 1 g 1n β 1 g g(a, b) = (α 1, α,..., α n ) 1 g g n β g n1 g n g nn β n = t a A Gb A. g(a, b) = g(b, a), a, b V g ij = g ji, i, j {1,,..., n} G = t G K = C G K = R G V = K n V E = (e 1, e,..., e n ) a V a = a E E n G g(a, b) := t agb, a, b V ( ) g : V V K G g maxima metrics.mac

62 6 /* metrics.mac */ load(eigen)$ geom(a, b, G) := block([ta], ta : conjugate(transpose(a)), ratsimp(ta. G. b) )$ geom G 57. (C1) load("metrics.mac"); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) metrics.mac (C) ( G : matrix([1, ) - %i], [ + %i, 3]); 1 i (D) i + 3 (C3) ( a :) matrix([s], [t]); s (D4) t (C5) ( b : ) matrix([x], [y]); x (D5) y (C6) geom(a, b, G); (D6) (3t + ( i) s) y + ((i + ) t + s) x (C7) geom(b, a, G); (D7) (3t + (i + ) s) y + (( i) t + s) x (C8) geom(a, a, G); (D8) 3t + 4st + s (C9) geom(b, b, G); (D9) 3y + 4xy + x (C10) kill(a,b); (D10) DONE (C11) ( a : ) matrix([1], []); 1 (D11) (C1) ( b : matrix([ ) - 3 * %i], [3 + %i]); 3i (D1) i + 3 (C13) geom(a, b, G); (D13) 41 6i (C14) geom(b, a, G); (D14) 6i + 41 (C15) geom(a, a, G); (D15) 1

63 (C16) geom(b, b, G); (D16) 77 maxima 63 G I : G = I =......, g ij = δ ij = { 1 (i = j) 0 (i j) g K n K = C C n K = R R n maxima innerproduct inprod 58. (C1) load("metrics.mac") Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) metrics.mac (C) a : transpose(matrix([s, t, u])); (D) (C3) s t u b : transpose(matrix([x, y, z])); (D5) x y z (C6) inprod(a, b); (D6) uz + ty + sx (C7) I : ident(3); (D7) (C8) geom(a, b, I); (D8) uz + ty + sx (C9) kill(a, b) (D9) DONE (C10) a : matrix([s, t, u]); (D10) ( s t u ) (C11) b : matrix([x, y, z]); (D11) ( x y z )

64 64 (C1) inprod(a, b); (D1) uz + ty + sx (C13) geom(a, b, I); (D13) uz + ty + sx (C14) kill(a, b); (D14) DONE (C15) a : [s, t, u]; (D18) [s, t, u] (C19) b : [x, y, z]; (D16) [x, y, z] (C17) inprod(a, b); (D19) uz + ty + sx (C0) geom(a, b, I); (D0) uz + ty + sx inprod load(eigen) g g(a, b) a, b := g(a, b) K n a a ( ) a a := a, a K = R R n a, b a b θ (0 θ π) cos θ := a, b a b V V V g a := g(a, b), cos θ := g(a, b) a b g V = K n G g absolute cosine anglcos metrics.mac absolute(a, G) := ratsimp(sqrt(geom(a, a, G)))$ anglcos(a, b, G) := block([denom], denom : absolute(a, G) * absolute(a, G),

65 maxima 65 )$ ratsimp(geom(a, b, G) / denom) 59. (C1) load("metrics.mac"); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) metrics.mac (C) I : ident(4); (D) (C3) len(a) := absolute(a, I); (D3) len (a) := absolute (a, I) (C4) ancos(a, b) := anglcos(a, b, I); (D4) ancos (a, b) := anglcos (a, b, I) (C5) a : transpose(matrix([1, -, 0, -1])); 1 (D5) 0 1 (C6) b : transpose(matrix([, 1, 3, -1])); (D6) (C7) len(a); (D7) 6 (C8) len(b); (D8) 15 (C9) (D9) (C10) (D10) ancos(a, b); inprod(a, b)/ (len(a) * len(b)); (C11) kill(a,b); (D11) DONE (C1) a : transpose(matrix([s, t, u, v])); s (D1) t u v (C13) b : transpose(matrix([w, x, y, z]));

66 66 (D13) w x y z (C14) len(a); (D14) v + u + t + s (C15) len(b); (D15) z + y + x + w (C16) inprod(a, b); (D16) vz + uy + tx + sw (C17) ancos(a, b); (D17) vz+uy+tx+sw v +u +t +s z +y +x +w V (K n ) V A B a b A a A b A a b B a B b B g A G A B G B g(a, b) t a A G A b A = t a B G B b B (= g(a, b)) A B P a B = P a A, A = BP b B = P b A t a B G B b B = t a At P G B P b A = t a A G A b A a b t P G B P = G A t P 1 G A P 1 = G B V g V g (V, g) (C n,, ) (R n,, ) R n E 1 O 1 M =... O 1

67 maxima 67 R n η (1, n 1) R 4 (1, 3) (R 4, η) R 1,3 a R 1,3 η η(a, a) η(a, a) < 0 a η(a, a) = 0 a η(a, a) > 0 a ( ) η(a, a) a := 0 η(a, a) a 1 a a, b R 1,3 b a a b µ cosh µ := η(a, b) a b maxima MinkMatrix mink tminknorm sminknorm minknorm hyperbolic cosine hypangcos MinkMatrix : ematrix(4, 4,, 1, 1) - ident(4)$ mink(a, b) := geom(a, b, MinkMatrix)$ tminknorm(a) := sqrt(mink(a,a))$ sminknorm(a) := sqrt(-mink(a,a))$ minknorm(a) := block( if mink(a, a) < 0 then sqrt(-mink(a, a)) else sqrt(mink(a, a)) )$ hypangcos(a, b) := ratsimp(mink(a, b) / (tminknorm(a) * tminknorm(b)))$ metrics.mac

68 (C1) load("metrics.mac"); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) metrics.mac (C) MinkMatrix; (D) (C3) a : transpose(matrix([s, t, u, v])); s (D3) t u v (C4) b : transpose(matrix([w, x, y, z])); w (D4) x y z (C5) mink(a, b); (D5) vz uy tx + sw (C6) tminknorm(a); (D6) v u t + s (C7) sminknorm(a); (D7) v + u + t s (C8) hypangcos(a, b); (D8) vz+uy+tx sw v u t +s z y x +w (C9) minknorm(a); MACSYMA was unable to evaluate the predicate: ERREXP 1 # 0: minknorm(a=matrix([s],[t],[u],[v]))(metrics.mac line 10) -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) (C10) kill(a, b); (D10) DONE (C11) a : transpose(matrix([5, 1, 0, -])); 5 (D11) 1 0 (C1) b : transpose(matrix([6,, -1, ])); 6 (D1) 1

69 (C13) mink(a, b); (D13) 3 (C14) mink(a, a); (D14) 0 (C15) mink(b, b); (D15) 7 (C16) minknorm(a); (D16) 5 (C17) minknorm(b); (D17) 3 3 (C18) hypangcos(a, b); (D18) maxima 69 (C19) c : transpose(matrix([1,, -1, 3])); 1 (D19) 1 3 (C0) mink(c, c); (D0) 13 (C1) sminknorm(c); (D1) 13 (C) minknorm(c); (D) Gram-Schmidt. (V, g) (a 1,..., a k ) (a 1,..., a k ) Gram- Schmidt (a 1,..., a k ) { g(a i, a i) 0 i {1,..., k} g(a i, a j) = 0 i j {1,..., k} g(v, w) = 0 v w (g ) v w (a 1,..., a k ) (a 1,..., a k ) {a 1,..., a k } = {a 1,..., a k} {a 1,..., a k } α 1 a α k a k = 0 i {1,..., k} a i g α i g(a i, a i) = 0

70 70 g(a i, a i) 0 α i = 0, i (a 1,..., a k ) {a 1,..., a k } {a 1,..., a k } (V, g) n (a 1,..., a k ) (k n) V k (g(a i, a i ) 0, i) (a 1,..., a k ) a 1 := a 1 (a 1,..., a l ) (l k 1) a l+1 a l+1 := a l+1 g(a l+1, a 1) g(a 1, a 1) a 1 g(a l+1, a l ) g(a l, a l ) a l i {1,..., l} g(a l+1, a i) = g(a l+1, a i) g(a l+1, a i) g(a i, a i ) g(a i, a i) = 0 {a 1,..., a l, a l+1} g(a l+1, a l+1 ) 0 (a 1,..., a l+1 ) (a 1,..., a k ) (a 1,..., a k ) (a 1,..., a k ) Gram-Schmidt g (a a i := 1 g(a i, a i )a i, i {1,..., k} 1,..., a k ) { g(a i, a i ) = 1 i {1,..., k} g(a i, a j ) = 0 i j {1,..., k} (a 1,..., a k ) (a 1,..., a k ) Gram-Schmidt R n C n Gram-Schmidt maxima maxima gramschmidt maxima gramschmidt load(eigeni);

71 maxima (C1) load(eigeni); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) /usr/local/share/maxima/5.9.0/share/matrix/eigeni.mac (C) gramschmidt([[,3],[1,-]]); (D) [ [, 3], [ 37, ]] (C3) a1 : [1,, 3]; (D3) [1,, 3] (C4) a : [4, 5, 6]; (D4) [4, 5, 6] (C5) a3 : [7, 8, -1]; (D5) [7, 8, 1] (C6) [ gramschmidt([a1, [ ] a, a3]); (D6) [1,, 3], 3, 3, 3, [ 5, 5, ] ] (C7) A : matrix([1,, 3], [4, 5, 6], [7, 8, -1]); (D7) (C8) [ gramschmidt(a); [ ] (D8) [1,, 3], 3, 3, 3, [ 5, 5, ] ] maxima gson.mac /* gson.mac */ load(eigeni)$ gso(a) := block([m,n,i,b,c,d], m : length(a), B : transpose(a), n : length(b), C : gramschmidt(b), D : zeromatrix(n, m), for i:1 thru n do D[i] : C[i], transpose(d) )$ gson(a) := block([m,n,i,b,c,d], m : length(a),

72 7 B : transpose(a), n : length(b), C : gramschmidt(b), D : zeromatrix(n, m), for i:1 thru n do D[i] : unitvector(c[i]), transpose(d) )$ gso gson gso Gram-Schmidt gson Gram-Schmidt maxima unitvector ( ) 6. (C1) load("gson.mac"); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) gson.mac (C) ( A : transpose(matrix([, ) -1], [-1, ])); 1 (D) 1 (C3) ( ratsimp(gso(a)); ) 3 (D3) 5 (C4) (D4) ratsimp(gson(a)); ) ( ] 5 (C5) [ a[1] : transpose(d4)[1]; (D5) 5, 1 (C6) [ a[] : transpose(d4)[]; 5 ] (D6) (C7) inprod(a[1], a[]); (D7) 0 (C8) A : transpose(matrix([1,, 3], [4, 5, 6], [7, 8, -1])); (D8) (C9) ratsimp(gso(a)); (D9) (C10) ratsimp(gson(a));

73 (D10) maxima (C11) B : transpose(matrix([1, 0, -, 1], [, 3, 3, -1], [0, 1, -1, -1])); (D11) (C1) rank(b); (D1) 3 (C13) ratsimp(gso(b)); (D13) (C14) (D14) ratsimp(gson(b)); gson A = (a 1,..., a n ) A = (a 1,..., a n) = gson(a) t A A = t A = t a 1a 1 t a a 1 t a 1. t a n t a 1a t a a t a t a.... 1a n a n.. t a na 1 t a na t a na n i, j {1,..., n} t a i a j = a i, a j = { 1 (i = j) 0 (i j) t A A = I n t A = (A ) 1 K = R A K = C A

74 (C1) load("gson.mac"); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) gson.mac (C) ( A : matrix([1, ) ], [3, 4]); 1 (D) 3 4 (C3) AA : ratsimp(gson(a)); ) (D3) (C4) (D4) (C5) (D5) ( transpose(aa); ) ( AA^^(-1); ( ) (C6) ( ratsimp(aa^^(-1) ) - transpose(aa)); 0 0 (D6) 0 0 (C7) ( ratsimp(transpose(aa) ). AA); 1 0 (D7) 0 1 (C8) B : matrix([1+%i,, -%i], [0, -%i, 1-*%i], [0, 0, +3*%i]); (D8) i + 1 i 0 i 1 i 0 0 3i + (C9) (D9) 3 (C10) (D10) (C11) (D11) (C1) (D1) (C13) rank(b); BB : ratsimp(gson(b)); i i 5 0 3i conjugate(transpose(bb)); 1 i i i 13 BB^^(-1); 0 0 i i i+ ratsimp(conjugate(transpose(bb)) - BB^^(-1));

75 (D17) maxima 75 (D13) (C14) ratsimp(conjugate(transpose(bb)). BB); (D14) (C15) ( C : matrix([1+%i, ) -%i], [3+*%i, 4*%i]); i + 1 i (D15) i + 3 4i (C16) rank(c); (D16) (C17) ( CC : ratsimp(gson(c)); i i 10 ) i+3 5 5i (C18) ( ratsimp(conjugate(transpose(cc)) ) - CC^^(-1)); 0 0 (D18) 0 0 (C19) ( ratsimp(conjugate(transpose(cc)) ). CC); 1 0 (D19) V V V V g (V, g) V (V, g) f : V V V f g g(f(a), f(b)) = g(a, b), a, b V f g V A f (A, A) F g A G V a b A a A, b A K n n = dim V f g A t (F a A )G(Ab A ) = t a A ( t F GA)b A = t a A Gb A a A b A t F GF = G F G V = K n, E

76 76 I n f (E, E) F I n t F I n F = t F F = I n K = R t F F = I n F K = C t F F = I n F K n f K = R K = C (V, g) V A = (a 1,..., a n ) { 1 (i = j) g(a i, a j ) = 0 (i j) V B = (b 1,... b n ) v V A B v A v B A B P v B = P v A P K R C e i, e j = g(a i, a j ) = g(b i, b j ), i, j P v A, P w A = v B, w B = g(v, w) = v A, w A v, w V t P P = I n 64. (C1) load("gson.mac"); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) gson.mac (C) A : ratsimp(gson(matrix([1,,3],[4,5,6],[7,8,0]))); (D) (C3) B : ratsimp(gson(matrix([1,0,1],[0,1,],[1,3,-1])));

77 (D3) maxima (C4) P : A^^(-1). B; (D4) (C5) (D5) (C6) (D6) (C7) (D7) (C8) (D8) (C9) (D9) (C10) (D10) ratsimp(transpose(p). P); ratsimp(transpose(a). A); ratsimp(transpose(b). B); ( C : ratsimp(gson(matrix([1+%i, ], [-%i, 3+*%i]))); i i 13 ) i 10 7i D : ratsimp(gson(matrix([1, ) %i], [1-%i, -%i]))); ( 1 3 3i i 1 3 3i Q : ratsimp(c. D); ( )i ( )i ( )i ( )i (C11) ( ratsimp(conjugate(transpose(q)) ). Q); 1 0 (D11) 0 1 (C1) ( ratsimp(conjugate(transpose(c)) ). C); 1 0 (D1) 0 1 (C13) ( ratsimp(conjugate(transpose(d)) ). D); 1 0 (D13) 0 1 A, B, P C, D, Q A t A A = t AA = I n = 1.

78 78 A 1 A A = t A A = t AA = I n = 1 A = ± (C1) load("gson.mac"); Warning - you are redefining the MACSYMA function EIGENVALUES Warning - you are redefining the MACSYMA function EIGENVECTORS (D1) gson.mac (C) A : ratsimp(gson(matrix([1,,3],[4,5,6],[7,8,0]))); (D) ratsimp(determinant(a)^); (C3) (D3) 1 (C4) B : ratsimp(gson(matrix([1,%i,-%i],[3+%i, 4*%i, ],[-1+%i, -3*%i, %i]))); (D4) (C5) (D5) 7i (C6) i i i i i i i i detb : ratsimp(determinant(b)); ratsimp(conjugate(detb)*detb); (D6) 1 ( ) a b A = c d t AA = I ( ) ( ) a c a b b d c d = ( ) a + c ab + cd ba + dc b + d = ( ) ( ) ( ) cos θ sin θ cos η sin η A = sin θ cos θ sin η cos η 1 1 θ ( )

79 maxima (C1) ( A : matrix([cos(th), ) -sin(th)], [sin(th), cos(th)]); cos th sin th (D1) sin th cos th (C) ( transpose(a). A; ) sin (D) th + cos th 0 0 sin th + cos th (C3) ( trigsimp(d); ) 1 0 (D3) 0 1 (C4) trigsimp(determinant(a)); (D4) 1 (C5) ( B : matrix([cos(phi), ) sin(phi)], [sin(phi), -cos(phi)]); cos ϕ sin ϕ (D5) sin ϕ cos ϕ (C6) ( transpose(b). B; ) sin (D6) ϕ + cos ϕ 0 0 sin ϕ + cos ϕ (C7) ( trigsimp(d6); ) 1 0 (D7) 0 1 (C8) trigsimp(determinant(b)); (D8) 1 (C9) ( T : matrix([0, ) 1], [1, 0]); 0 1 (D9) 1 0 (C10) ( C : B. T; ) sin ϕ cos ϕ (D10) cos ϕ sin ϕ (C11) ( trigsimp(transpose(c) ). C); 1 0 (D11) 0 1 (C1) trigsimp(determinant(c)); (D1) 1 maxima trigsimp R n η R 1,n 1 = (R n, η) (n = 4 ) η E J J =

80 80 R 1,n 1 f η f (E, E) F F t F JF = J (E, E) R 1,n 1 R 1,n 1 A = (a 1,..., a n ) 1 (i = j = 1) η(a i, a j ) = 1 (i = j 1) 0 (i j) A R 1,n 1 R 1,n 1 A = (a 1,..., a n ) B = (b 1,..., b n ) A B P t P JP = J P v B = P v A F F J = t F J F = t F JF = J J = 0 F = 1 F = ±1 ( ) a b A = c d ( ) ( ) ( ) ( ) a c 1 0 a b 1 0 = b d 0 1 c d 0 1 ( ) a c ab cd ab cd b d = A ( ) ( ) cosh µ sinh µ cosh µ sinh µ,, sinh µ cosh µ sinh µ cosh µ ( ) ( cosh µ sinh µ sinh µ cosh µ v = tanh µ 1 cosh µ =, sinh µ = v 1 v 1 v ), ( cosh µ sinh µ ) sinh µ cosh µ