/Users/yamada/Documents/webPage/public_html/kkk/10 線形代数

Size: px

Start display at page:

Download "/Users/yamada/Documents/webPage/public_html/kkk/10 線形代数"

くにひとかつま
4 years ago
Views:

1 8 Mathematica In[]:= 8, 2, 3< Out[]= 8, 2, 3< In[2]:= 88, 2, 3<, 84, 5, 6<< Out[2]= 88, 2, 3<, 84, 5, 6<< 2 3 MatrixQ MatrixQ In[3]:= Out[3]= In[4]:= Out[4]= MatrixQ@88, 2, 3<, 84, 5, 6<<D True MatrixQ@88, 2, 3<, 84, 5<<D False MatrixForm MatrixForm In[5]:= MatrixForm@88, 2, 3<, 84, 5, 6<<D Out[5]//MatrixForm= J N In[6]:= MatrixForm@8, 2, 3<D Out[6]//MatrixForm= i y 2 j k 3 {

2 96 0 MatrixForm {} In[7]:= MatrixForm@88, 2, 3 <<D Out[7]//MatrixForm= H 2 3 L Dimensions Dimensions[mat] mat In[8]:= Dimensions@88, 2, 3<, 84, 5, 6<<D Out[8]= 82, 3< Array, Table DiagonalMatrix, IdentityMatrix Array In[9]:= Array@a, 82, 3<D Out[9]= 88a@, D, a@, 2D, a@, 3D<, 8a@2, D, a@2, 2D, a@2, 3D<< i, j a[i, j] 2µ3 Table i, j i+j 2µ3 In[0]:= Table@i + j, 8i, 2<, 8j, 3<D Out[0]= 882, 3, 4<, 83, 4, 5<< In[]:= MatrixForm@%D Out[]//MatrixForm= J N In[2]:= Table@Random@D, 82<, 83<D Out[2]= , , <, , , << 0

3 0 97 In[3]:= 82<, 83<D Out[3]= 880, 0, 0<, 80, 0, 0<< i, j a i j = i - j 3 µ 3 DiagonalMatrix 0 DiagonalMatrix DiagonalMatrix[{a, a2,...}] a, a2,... In[4]:= Out[4]= In[5]:= DiagonalMatrix@8a, b, c<d 88a, 0, 0<, 80, b, 0<, 80, 0, c<< MatrixForm@%D Out[5]//MatrixForm= i a 0 0 y 0 b 0 j k 0 0 c { IdentityMatrix IdentityMatrix IdentityMatrix[n] nµn In[6]:= IdentityMatrix@3D Out[6]= 88, 0, 0<, 80,, 0<, 80, 0, << In[7]:= MatrixForm@%D Out[7]//MatrixForm= i 0 0 y 0 0 j k 0 0 { A t A Transpose Transpose[mat] mat In[]:= MatrixForm@mat = Array@a, 83, 4<DD Out[]//MatrixForm= i a@, D a@, 2D a@, 3D a@, 4D y a@2, D a@2, 2D a@2, 3D a@2, 4D j k a@3, D a@3, 2D a@3, 3D a@3, 4D {

4 98 0 In[9]:= MatrixForm@Transpose@matDD Out[9]//MatrixForm= i a@, D a@2, D a@3, D y a@, 2D a@2, 2D a@3, 2D a@, 3D a@2, 3D a@3, 3D j k a@, 4D a@2, 4D a@3, 4D { Part mat In[24]:= MatrixForm@matD Out[24]//MatrixForm= i a@, D a@, 2D a@, 3D a@, 4D y a@2, D a@2, 2D a@2, 3D a@2, 4D j k a@3, D a@3, 2D a@3, 3D a@3, 4D { Part[mat, i, j] mat[[i, j]] mat i, j In[25]:= Out[25]= mat@@, 2DD a@, 2D Part[mat, {i, i2,...}, {j, j2,...}] mat[[{i, i2,...}, {j, j2,...}]] i, i2,... j, j2,... mat In[26]:= Part@mat, 8, 3<, 8, 2, 4<D Out[26]= In[27]:= 88a@, D, a@, 2D, a@, 4D<, 8a@3, D, a@3, 2D, a@3, 4D<< MatrixForm@%D Out[27]//MatrixForm= a@, D a@, 2D a@, 4D J a@3, D a@3, 2D a@3, 4D N Part[mat, i] mat[[i]] mat i In[28]:= mat@@2dd Out[28]= 8a@2, D, a@2, 2D, a@2, 3D, a@2, 4D<

5 0 99 Transpose[mat][[j]] mat j In[29]:= Transpose@matD@@2DD Out[29]= 8a@, 2D, a@2, 2D, a@3, 2D< +, - * " " In[30]:= 8, 2, 3< + 84, 5, 6< Out[30]= 85, 7, 9< In[3]:= 85, 5, 5 < - 8, 2, 3< Out[3]= 84, 3, 2< In[32]:= 88, 2<, 83, 4<< + 885, 6<, 87, 8<< Out[32]= 886, 8<, 80, 2<< In[33]:= 3 * 8, 2, 3< Out[33]= 83, 6, 9< In[34]:= 3 88, 2<, 83, 4<< Out[34]= 883, 6<, 89, 2<< {, 2, 3} + 4 Mathematica {, 2, 3} 4 {+4, 2+4, 3+4} In[35]:= 8, 2, 3< + 4 Out[35]= 85, 6, 7< "." Dot

6 00 0 Dot[mat, mat2] mat.mat2 mat mat2 l µ m Ha i j L m µ n Hb i j L l µ n Hc i j L c i j = m k= a i k b k j J NJ 0 2 N=J N In[36]:= 88, 2<, 82, 2<<. 88,, 0<, 82,, << Out[36]= 885, 3, 2<, 86, 4, 2<< J NJ N In[37]:= 88, 2, 3<, 82, 3, 4<<. 88, 2<, 83, 4<< Dot::dotsh : Tensors 88, 2, 3<, 82, 3, 4<< and 88, 2<, 83, 4<< have incompatible shapes. Out[37]= 88, 2, 3<, 82, 3, 4<<.88, 2<, 83, 4<< Dot[mat, vec] mat.vec mat vec m µ n Ha i j L n Hb i L m Hc i L n c i = k= a i k b k J 2 0 i y 2 0 N j =J 3 4 N k 2 { In[38]:= 88, 2, 0<, 82, 0, <<. 8,, 2< Out[38]= 83, 4< Dot[vec, mat] vec.mat vec mat m Hb i L m µ n Ha i j L n Hc i L c i = m k= b k a k i H 2 LJ N=H 4 4 L In[39]:= 82, <.88, 2, 0<, 82, 0, << Out[39]= 84, 4, < Mathematica

7 0 0 a i, i 2,..., i k i, i 2,..., i k m µ m 2 µ µ m k a b j, j 2,..., j l j, j 2,..., j l n µ n 2 µ µ n l b a.b a i k b j m µ m 2 µ µ m k- µ n 2 µ µ n l {{, 2, 0}, {2, 0, }}. {,, 2} 2µ {2, }.{{, 2, 0}, {2, 0, }} 2 2µ3 3 3 Dot[vec, vec2] vec.vec2 vec vec2 m Ha i L Hb i L m i= a i b i In[40]:= 8, 2, 3<. 82, 3, 4< Out[40]= 20 v Sqrt[v.v] In[4]:= Sqrt@8, 2, 3<.8, 2, 3<D Out[4]= è!!!!!! 4 "." "*" " " In[39]:= 8a, b, c< * 8x, y, < Out[39]= 8a x, b y, c < mata, matb, matc i, j a[i, j], b[i, j], c[i, j] 2 µ 2 (mata. matb). matc mata. (matb. matc) A A n MatrixPower MatrixPower[mat, n] mat n mat.mat..mat

8 02 0 In[45]:= MatrixForm@mat = 88, <, 82, <<D Out[45]//MatrixForm= J 2 N mat {{, }, {2, }} MatrixForm mat 3 In[46]:= MatrixForm@MatrixPower@mat, 3DD Out[46]//MatrixForm= J N MatrixPower n In[48]:= MatrixForm@MatrixPower@mat, ndd Out[48]//MatrixForm= i è!!! ÅÅÅ H - 2 L n + è!!! ÅÅÅ H + 2 L n - I-è!!!! 2 M ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ n 2 2 j k - I-è!!!! 2 M ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ n è!!!! + I+è!!!! 2 M n è!!!! 2 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 2 è!!!! 2 è!!! ÅÅÅ H - 2 L n + ÅÅÅ I+è!!!! 2 M ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ n 2 è!!!! 2 H + è!!! 2 L n y { mat mat^n In[40]:= 88a, b<, 8c, d<< ^ n Out[40]= 88a n, b n <, 8c n, d n << J N n MatrixPower 2 0 A A - Inverse Inverse[mat] mat In[4]:= Inverse@88, 2<, 83, 4<<D Out[4]= 98-2, <, 9 ÅÅÅÅ 3 2, - ÅÅÅÅ 2 == In[5]:= 88, 2<, 83, 4<<.% Out[5]= 88, 0<, 80, <<

9 0 03 In[54]:= 2<, 82, 4<<D Inverse::sing : Matrix 88, 2<, 82, 4<< is singular. Out[54]= Inverse@88, 2<, 82, 4<<D i 2 y i 2 y j k 0 2 { j k 0 2 { A dethal Det Det Det[mat] mat In[55]:= Out[55]= In[56]:= Out[56]= Det@88a, b<, 8c, d<<d -b c + a d Det@88a, b, c<, 8d, e, f<, 8g, h, i<<d -c e g + b f g + c d h - a f h - b d i + a e i 0 In[57]:= Det@88, 2<, 82, 4<<D Out[57]= 0 mata, matb i, j a[i, j], b[i, j] 2 µ 2 mata. matb Factor mata matb A 0 x A x = l x l A x A Eigenvalues, Eigenvectors Eigensystem mat In[58]:= MatrixForm@matD Out[58]//MatrixForm= J 2 N

10 04 0 Eigenvalues Eigenvalues[mat] mat In[59]:= Out[59]= eval = Eigenvalues@matD 8 - è!!! 2, + è!!! 2 < Eigenvectors Eigenvectors[mat] In[60]:= evec = Eigenvectors@matD Out[60]= 99- ÅÅÅÅÅÅÅÅÅ è!!! 2, =, 9 ÅÅÅÅÅÅÅÅÅ è!!! 2, == mat In[6]:= mat.evec@@dd Out[6]= 9 - ÅÅÅÅÅÅÅÅÅ è!!! 2, - è!!! 2 = In[62]:= Out[62]= evec@@dd eval@@dd 9- - è!!! 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!, - è!!! 2 = 2 Simplify In[63]:= Simplify@%D Out[63]= 9 - ÅÅÅÅÅÅÅÅÅ è!!! 2, - è!!! 2 = Eigensystem Eigensystem[mat] In[64]:= Out[64]= Eigensystem@matD 98 - è!!! 2, + è!!! 2 <, 99- ÅÅÅÅÅÅÅÅÅ è!!! 2, =, 9 ÅÅÅÅÅÅÅÅÅ è!!! 2, === A A x = b LinearSolve, NullSpace

11 0 05 LinearSolve LinearSolve[m, b] m.x == b x In[65]:= MatrixForm@m = 88, 2, 3, 4<, 8,,, <<D Out[65]//MatrixForm= J N In[66]:= MatrixForm@b = 82, 3<D Out[66]//MatrixForm= J 2 3 N x0 J Nx = J 2 3 N In[67]:= MatrixForm@x0 = LinearSolve@m, bdd Out[67]//MatrixForm= i 4 y - 0 j k 0 { m.x0 == b In[68]:= MatrixForm@m.x0D Out[68]//MatrixForm= J 2 3 N NullSpace NullSpace[m] m m.x == 0 m m In[69]:= ns = NullSpace@mD Out[69]= 882, -3, 0, <, 8, -2,, 0<< m.x == 0 In[70]:= m.ns@@dd Out[70]= 80, 0< In[7]:= m.ns@@2dd Out[7]= 80, 0<

12 06 0 m.x == b x LinearSolve x0 NullSpace m ns s, t In[72]:= MatrixForm@x = x0 + s ns@@dd + t ns@@2ddd Out[72]//MatrixForm= i s + t y s - 2 t t j k s { s ns[[]] + t ns[[2]] {s, t}.ns m.x == b m.x - b == 0 In[73]:= Simplify@m.x - bd Out[73]= 80, 0< [8-] i, j i 2 - i j + 3 j 2 3µ3 mat [8-2] [8-] mat [8-3] 3µ3 A, B t HA BL = t B t A Mathematica [8-4] q cos q = ÅÅÅÅÅÅÅÅÅÅÅ uÿv u ÿ v» u»»u»»v» u (, 3, ), (-2,, 3) [8-5] J NJ x y N=J N s, t 2

ParametricPlot [5] In[5]:= Out[5]= m1.v1 axby, cxdy [6] pr In[6]:= Out[6]= pr = m1.m 3ab, ab,3cd, cd [7] In[7]:= Out[7]//MatrixForm= pr //Matri

ParametricPlot [5] In[5]:= Out[5]= m1.v1 axby, cxdy [6] pr In[6]:= Out[6]= pr = m1.m 3ab, ab,3cd, cd [7] In[7]:= Out[7]//MatrixForm= pr //Matri Chapter 6 1 ParametricPlot 3 ParametricPlot ParametricPlot3D 6.1 [1] *1 In[1]:= v1 = {x, y}; v = {z, w}; [] In[]:= m1 = {{a, b}, {c, d}}; m = {{3, }, {1, }}; [3] 3 3.3 In[3]:= Out[3]//MatrixForm= [] m1