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length(a) a:

>> syms x y z

>> length([x y z])

ans =

prod(a) a:

>> syms x y z

>> prod([x y z])

ans =

x*y*z

sum(a) a:

>> syms x y z

>> sum([x y z])

ans =

x + y + z

mean(a) a:

>> syms x y z

>> mean([x y z])

ans =

x/3 + y/3 + z/3

dot(a,b) a b:

>> a1=sym('[x1 y1 z1]');

>> a2=sym('[x2 y2 z2]');

>> dot(a1,a2)

ans =

x2*conj(x1) + y2*conj(y1) + z2*conj(z1)

cross(a,b) a b:

>> a1=sym('[x1 y1 z1]');

>> a2=sym('[x2 y2 z2]');

>> cross(a1,a2)

ans =

[ y1*z2 - y2*z1, x2*z1 - x1*z2, x1*y2 - x2*y1]

sort(a) a:

>> syms a b c d e f g h i j k

>> sort([k d j a i b g c e h f])

ans =

[ a, b, c, d, e, f, g, h, i, j, k]

.

diag(a [, k]) a k- . diag(A [, k]), A , -, k- A. :

>> syms a b c d e f g h i j k

>> diag([a b c])

ans =

[ a, 0, 0]

[ 0, b, 0]

[ 0, 0, c]

>> diag([f,h],2)

ans =

[ 0, 0, f, 0]

[ 0, 0, 0, h]

[ 0, 0, 0, 0]

[ 0, 0, 0, 0]

>> diag([i,j],-2)

ans =

[ 0, 0, 0, 0]

[ 0, 0, 0, 0]

[ i, 0, 0, 0]

[ 0, j, 0, 0]

>> M=[a b c; d e f; h i k]

M =

[ a, b, c]

[ d, e, f]

[ h, i, k]

>> diag(M)

ans =

a

e

k

repmat(A, n [, m]) , n n n m A, A , , A. :

>> syms a b c d e f g h i

>> A=[a b c; d e f; g h i]

A =

[ a, b, c]

[ d, e, f]

[ g, h, i]

>> repmat(A,2)

ans =

[ a, b, c, a, b, c]

[ d, e, f, d, e, f]

[ g, h, i, g, h, i]

[ a, b, c, a, b, c]

[ d, e, f, d, e, f]

[ g, h, i, g, h, i]

reshape(A, m, n) m n, A ( A m n , ):

>> syms a b c d e f

>> A=[a b c; d e f]

A =

[ a, b, c]

[ d, e, f]

>> reshape(A,3,2)

ans =

[ a, e]

[ d, c]

[ b, f]

cat(n, A, B [, C,...]) A B A, B, C .. :

>> syms a b c d e f g h

>> A=[a b; c d]

A =

[ a, b]

[ c, d]

>> B=[e f; g h]

B =

[ e, f]

[ g, h]

>> cat(1,A,B)

ans =

[ a, b]

[ c, d]

[ e, f]

[ g, h]

>> cat(2,A,B)

ans =

[ a, b, e, f]

[ c, d, g, h]

rot90(A [, k]) A 90 90k, k . :

>> syms a b c d e f g h i

>> A=[a b c; d e f; g h i]

A =

[ a, b, c]

[ d, e, f]

[ g, h, i]

>> rot90(A)

ans =

[ c, f, i]

[ b, e, h]

[ a, d, g]

tril(A [, k]) A , k- .
triu(A [, k]) A , k- . :

>> syms a b c d e f g h i

>> A=[a b c; d e f; g h i]

A =

[ a, b, c]

[ d, e, f]

[ g, h, i]

>> tril(A)

ans =

[ a, 0, 0]

[ d, e, 0]

[ g, h, i]

>> triu(A)

ans =

[ a, b, c]

[ 0, e, f]

[ 0, 0, i]

size(A) A, [n m], n , m . :

>> syms a b c d e f

>> A=[a b c; d e f]

A =

[ a, b, c]

[ d, e, f]

>> size(A)

ans =

2 3

prod(A [, k]) - ( -), k, ( ) A. k , . , prod(prod(A)) A. :

>> syms a b c d e f g h i

>> A=[a b c; d e f; g h i]

A =

[ a, b, c]

[ d, e, f]

[ g, h, i]

>> prod(A)

ans =

[ a*d*g, b*e*h, c*f*i]

>> prod(prod(A))

ans =

a*b*c*d*e*f*g*h*i

sum(A [, k]) - ( -), k, ( ) A. k , . sum(sum(A)) A. :

>> syms a b c d e f g h i

>> A=[a b c; d e f; g h i];

>> sum(A)

ans =

[ a + d + g, b + e + h, c + f + i]

>> sum(sum(A))

ans =

a + b + c + d + e + f + g + h + i

mean(A [, k]) - ( -), k, ( ) A. k , . sum(sum(A)) A. :

>> syms a b c d e f g h i

>> A=[a b c; d e f; g h i];

>> mean(A)

ans =

[ a/3 + d/3 + g/3, b/3 + e/3 + h/3, c/3 + f/3 + i/3]

>> mean(mean(A))

ans =

a/9 + b/9 + c/9 + d/9 + e/9 + f/9 + g/9 + h/9 + i/9

sort(A) , A, . :

>> syms a b c d e f g h i

>> A=[d a i; b g c; e h f]

A =

[ d, a, i]

[ b, g, c]

[ e, h, f]

>> sort(A)

ans =

[ b, a, c]

[ d, g, f]

[ e, h, i]

.

det(A) A, ..

>> syms a b c d e f g h i

>> A=[a b; c d]

A =

[ a, b]

[ c, d]

>> det(A)

ans =

a*d - b*c

>> B=[a b c; d e f; g h i]

B =

[ a, b, c]

[ d, e, f]

[ g, h, i]

>> det(B)

ans =

a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g

trace(A) A, .. . :

>> syms a b c d e f g h i

>> A=[a b c; d e f; g h i]

A =

[ a, b, c]

[ d, e, f]

[ g, h, i]

>> trace(A)

ans =

a + e + i

inv(A) , A. :

>> syms a b c d

>> A=[a b; c d]

A =

[ a, b]

[ c, d]

>> inv(A)

ans =

[ d/(a*d - b*c), -b/(a*d - b*c)]

[ -c/(a*d - b*c), a/(a*d - b*c)]

eig(A) A. [M,D]=eig(A) M, A, D, A. eig(A,B), A B , . :

>> syms x

>> A=[1 -1; a 2]

A =

[ 1, -1]

[ a, 2]

>> [M,D]=eig(A)

M =

[-((1-4*a)^(1/2)/2-3/2)/a-2/a, ((1-4*a)^(1/2)/2+3/2)/a-2/a]

[ 1, 1]

D =

[ 3/2 - (1 - 4*a)^(1/2)/2, 0]

[ 0, (1 - 4*a)^(1/2)/2 + 3/2]

charpoly(A) - A. :

>> syms x

>> A=[1 -1; a 2];

>> charpoly(A)

ans =

[ 1, -3, a + 2]

.

, .

:

>> syms alpha;

>> A=[cos(alpha) sin(alpha); -sin(alpha) cos(alpha)]

A =

[ cos(alpha), sin(alpha)]

[ -sin(alpha), cos(alpha)]

>> B=A^2

B =

[ cos(alpha)^2 - sin(alpha)^2, 2*cos(alpha)*sin(alpha)]

[ -2*cos(alpha)*sin(alpha), cos(alpha)^2 - sin(alpha)^2]

>> B=simplify(B)

B =

[ cos(2*alpha), sin(2*alpha)]

[ -sin(2*alpha), cos(2*alpha)]

>> I=simplify(A.'*A)

I =

[ 1, 0]

[ 0, 1]

.

:

>> A=sym([12 32 66 116; -25 -76 -164 -294;...

21 66 143 256; -6 -19 -41 -73])

A =

[ 12, 32, 66, 116]

[ -25, -76, -164, -294]

[ 21, 66, 143, 256]

[ -6, -19, -41, -73]

>> [V,J]=jordan(A)

V =

[ 4, -2, 4, 3]

[ -6, 8, -11, -8]

[ 4, -7, 10, 7]

[ -1, 2, -3, -2]

J =

[ 1, 1, 0, 0]

[ 0, 1, 0, 0]

[ 0, 0, 2, 1]

[ 0, 0, 0, 2]

>> V*J*inv(V)

ans =

[ 12, 32, 66, 116]

[ -25, -76, -164, -294]

[ 21, 66, 143, 256]

[ -6, -19, -41, -73]

.

:

>> A=sym([12 32 66 116; -25 -76 -164 -294;...

21 66 143 256; -6 -19 -41 -73])

A =

[ 12, 32, 66, 116]

[ -25, -76, -164, -294]

[ 21, 66, 143, 256]

[ -6, -19, -41, -73]

>> [U,S,V]=svd(A);

>> double(U)

ans =

0.2830 0.9472 -0.1142 -0.0981

-0.7109 0.1073 -0.6415 -0.2675

0.6190 -0.3002 -0.5293 -0.4965

-0.1768 0.0334 0.5434 -0.8199

>> double(S)

ans =

486.7163 0 0 0

0 2.9955 0 0

0 0 0.4843 0

0 0 0 0.0057

>> double(V)

ans =

0.0724 0.7272 0.6016 0.3225

0.2205 0.5692 -0.3280 -0.7210

0.4747 0.2049 -0.6208 0.5893

0.8490 -0.3244 0.3810 -0.1698

>> double(U*S*V'-A)

ans =

1.0e-37 *

0.0588 0.1175 0 0.2351

0 0 0 0

0 0 0 0

0 0 0 -0.2351





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