format compact
% Priklad 1
Ar=-5+round(10*rand(3,5))
Ar =
5 0 0 -1 4
-3 4 -5 1 2
1 3 3 3 -3
Ai=-5+round(10*rand(3,5))
Ai =
-1 -1 -1 -4 1
4 4 3 -3 -2
4 -4 -5 -3 -3
Br=-5+round(10*rand(3,5))
Br =
-5 4 3 2 2
2 0 0 3 -1
-1 -1 -3 -5 3
Bi=-5+round(10*rand(3,5))
Bi =
0 -2 2 -3 4
2 -3 -2 2 4
-1 -3 0 -1 1
A=Ar+i*Ai
A =
Columns 1 through 4
5.0000 - 1.0000i 0 - 1.0000i 0 - 1.0000i -1.0000 - 4.0000i
-3.0000 + 4.0000i 4.0000 + 4.0000i -5.0000 + 3.0000i 1.0000 - 3.0000i
1.0000 + 4.0000i 3.0000 - 4.0000i 3.0000 - 5.0000i 3.0000 - 3.0000i
Column 5
4.0000 + 1.0000i
2.0000 - 2.0000i
-3.0000 - 3.0000i
B=Br+i*Bi
B =
Columns 1 through 4
-5.0000 4.0000 - 2.0000i 3.0000 + 2.0000i 2.0000 - 3.0000i
2.0000 + 2.0000i 0 - 3.0000i 0 - 2.0000i 3.0000 + 2.0000i
-1.0000 - 1.0000i -1.0000 - 3.0000i -3.0000 -5.0000 - 1.0000i
Column 5
2.0000 + 4.0000i
-1.0000 + 4.0000i
3.0000 + 1.0000i
C=B.'
C =
-5.0000 2.0000 + 2.0000i -1.0000 - 1.0000i
4.0000 - 2.0000i 0 - 3.0000i -1.0000 - 3.0000i
3.0000 + 2.0000i 0 - 2.0000i -3.0000
2.0000 - 3.0000i 3.0000 + 2.0000i -5.0000 - 1.0000i
2.0000 + 4.0000i -1.0000 + 4.0000i 3.0000 + 1.0000i
D=B'
D =
-5.0000 2.0000 - 2.0000i -1.0000 + 1.0000i
4.0000 + 2.0000i 0 + 3.0000i -1.0000 + 3.0000i
3.0000 - 2.0000i 0 + 2.0000i -3.0000
2.0000 + 3.0000i 3.0000 - 2.0000i -5.0000 + 1.0000i
2.0000 - 4.0000i -1.0000 - 4.0000i 3.0000 - 1.0000i
% Priklad 3
Bs=B(1:3,1:3)
Bs =
-5.0000 4.0000 - 2.0000i 3.0000 + 2.0000i
2.0000 + 2.0000i 0 - 3.0000i 0 - 2.0000i
-1.0000 - 1.0000i -1.0000 - 3.0000i -3.0000
Bs'*Bs
ans =
35.0000 -22.0000 + 6.0000i -16.0000 -17.0000i
-22.0000 - 6.0000i 39.0000 17.0000 + 5.0000i
-16.0000 +17.0000i 17.0000 - 5.0000i 26.0000
% Matice neni unitarni, nebot Bs'*Bs neni jednotkova matice
A+B
ans =
Columns 1 through 4
0 - 1.0000i 4.0000 - 3.0000i 3.0000 + 1.0000i 1.0000 - 7.0000i
-1.0000 + 6.0000i 4.0000 + 1.0000i -5.0000 + 1.0000i 4.0000 - 1.0000i
0 + 3.0000i 2.0000 - 7.0000i 0 - 5.0000i -2.0000 - 4.0000i
Column 5
6.0000 + 5.0000i
1.0000 + 2.0000i
0 - 2.0000i
A.*B
ans =
Columns 1 through 4
-25.0000 + 5.0000i -2.0000 - 4.0000i 2.0000 - 3.0000i -14.0000 - 5.0000i
-14.0000 + 2.0000i 12.0000 -12.0000i 6.0000 +10.0000i 9.0000 - 7.0000i
3.0000 - 5.0000i -15.0000 - 5.0000i -9.0000 +15.0000i -18.0000 +12.0000i
Column 5
4.0000 +18.0000i
6.0000 +10.0000i
-6.0000 -12.0000i
A./B
ans =
Columns 1 through 4
-1.0000 + 0.2000i 0.1000 - 0.2000i -0.1538 - 0.2308i 0.7692 - 0.8462i
0.2500 + 1.7500i -1.3333 + 1.3333i -1.5000 - 2.5000i -0.2308 - 0.8462i
-2.5000 - 1.5000i 0.9000 + 1.3000i -1.0000 + 1.6667i -0.4615 + 0.6923i
Column 5
0.6000 - 0.7000i
-0.5882 - 0.3529i
-1.2000 - 0.6000i
A*C
ans =
-35.0000 +11.0000i 4.0000 + 9.0000i 3.0000 +28.0000i
23.0000 -18.0000i 19.0000 + 3.0000i 30.0000 -16.0000i
21.0000 -84.0000i 2.0000 -17.0000i -45.0000 + 5.0000i
% Priklad 5
E=A(:,1:3)*C(1:3,:)
E =
-25.0000 - 2.0000i 7.0000 + 8.0000i -9.0000
18.0000 -13.0000i 4.0000 30.0000 -26.0000i
18.0000 -51.0000i -28.0000 - 5.0000i -21.0000 + 5.0000i
det(E)
ans =
-1.9200e+003 -1.2435e+004i
det(A(:,1:3))*det(C(1:3,:))
ans =
-1.9200e+003 -1.2435e+004i
system_dependent('setprintcolorchoice', -1)
; ; ; ; ; ;
disp(get(0, 'Echo'))
off
set(0, 'Echo', 'off')
set(0, 'Format', 'short')
set(0, 'FormatSpacing', 'compact')
feature('EightyColumns', 0);
system_dependent('TabCompletion', 100);
% Priklad 6
help eig
EIG Eigenvalues and eigenvectors.
E = EIG(X) is a vector containing the eigenvalues of a square
matrix X.
[V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a
full matrix V whose columns are the corresponding eigenvectors so
that X*V = V*D.
[V,D] = EIG(X,'nobalance') performs the computation with balancing
disabled, which sometimes gives more accurate results for certain
problems with unusual scaling.
E = EIG(A,B) is a vector containing the generalized eigenvalues
of square matrices A and B.
[V,D] = EIG(A,B) produces a diagonal matrix D of generalized
eigenvalues and a full matrix V whose columns are the
corresponding eigenvectors so that A*V = B*V*D.
EIG(A,B,'chol') is the same as EIG(A,B) for symmetric A and symmetric
positive definite B. It computes the generalized eigenvalues of A and B
using the Cholesky factorization of B.
EIG(A,B,'qz') ignores the symmetry of A and B and uses the QZ algorithm.
In general, the two algortihms return the same result, however using the
QZ algorithm may be more stable for certain problems.
The flag is ignored when A and B are not symmetric.
See also CONDEIG, EIGS.
Overloaded methods
help lti/eig.m
help sym/eig.m
lambda=eig(E)
lambda =
-40.7966 -30.5895i
3.8080 - 5.0586i
-5.0114 +38.6481i
prod(lambda)
ans =
-1.9200e+003 -1.2435e+004i
det(E)
ans =
-1.9200e+003 -1.2435e+004i
sum(lambda)
ans =
-42.0000 + 3.0000i
trace(E)
ans =
-42.0000 + 3.0000i
% Priklad 7
[m,n]=size(A)
m =
3
n =
5
[A;(1:m)*A]
ans =
Columns 1 through 4
5.0000 - 1.0000i 0 - 1.0000i 0 - 1.0000i -1.0000 - 4.0000i
-3.0000 + 4.0000i 4.0000 + 4.0000i -5.0000 + 3.0000i 1.0000 - 3.0000i
1.0000 + 4.0000i 3.0000 - 4.0000i 3.0000 - 5.0000i 3.0000 - 3.0000i
2.0000 +19.0000i 17.0000 - 5.0000i -1.0000 -10.0000i 10.0000 -19.0000i
Column 5
4.0000 + 1.0000i
2.0000 - 2.0000i
-3.0000 - 3.0000i
-1.0000 -12.0000i
% Priklad 8
x=B(:,1)
x =
-5.0000
2.0000 + 2.0000i
-1.0000 - 1.0000i
u=A(:,1)
u =
5.0000 - 1.0000i
-3.0000 + 4.0000i
1.0000 + 4.0000i
xp = (dot(u,x)/dot(u,u)).*u
xp =
-2.2941 - 0.7647i
2.1765 - 0.9412i
0.5294 - 1.8824i
dot(x-xp,u)
ans =
0
dot(u,u)
ans =
68
norm(u)^2
ans =
68
% Priklad 9
w=cross(x-xp,u)
w =
-13.0000 +11.0000i
-1.0000 +16.0000i
3.0000 -28.0000i
dot(w,u)
ans =
-1.1800e+002 +4.2000e+001i
dot(w,x-xp)
ans =
61.5294 -20.4706i
help cross
CROSS Vector cross product.
C = CROSS(A,B) returns the cross product of the vectors
A and B. That is, C = A x B. A and B must be 3 element
vectors.
C = CROSS(A,B) returns the cross product of A and B along the
first dimension of length 3.
C = CROSS(A,B,DIM), where A and B are N-D arrays, returns the cross
product of vectors in the dimension DIM of A and B. A and B must
have the same size, and both SIZE(A,DIM) and SIZE(B,DIM) must be 3.
See also DOT.
w=cross(u,x-xp)
w =
13.0000 -11.0000i
1.0000 -16.0000i
-3.0000 +28.0000i
w=cross(conj(u),x-xp)
w =
-3.4706 - 0.1765i
8.8824 + 8.7059i
-15.0000 + 6.0000i
dot(w,x-xp)
ans =
61.5294 +20.4706i
A
A =
Columns 1 through 4
5.0000 - 1.0000i 0 - 1.0000i 0 - 1.0000i -1.0000 - 4.0000i
-3.0000 + 4.0000i 4.0000 + 4.0000i -5.0000 + 3.0000i 1.0000 - 3.0000i
1.0000 + 4.0000i 3.0000 - 4.0000i 3.0000 - 5.0000i 3.0000 - 3.0000i
Column 5
4.0000 + 1.0000i
2.0000 - 2.0000i
-3.0000 - 3.0000i
kron(A,ones(3,2))
ans =
Columns 1 through 4
5.0000 - 1.0000i 5.0000 - 1.0000i 0 - 1.0000i 0 - 1.0000i
5.0000 - 1.0000i 5.0000 - 1.0000i 0 - 1.0000i 0 - 1.0000i
5.0000 - 1.0000i 5.0000 - 1.0000i 0 - 1.0000i 0 - 1.0000i
-3.0000 + 4.0000i -3.0000 + 4.0000i 4.0000 + 4.0000i 4.0000 + 4.0000i
-3.0000 + 4.0000i -3.0000 + 4.0000i 4.0000 + 4.0000i 4.0000 + 4.0000i
-3.0000 + 4.0000i -3.0000 + 4.0000i 4.0000 + 4.0000i 4.0000 + 4.0000i
1.0000 + 4.0000i 1.0000 + 4.0000i 3.0000 - 4.0000i 3.0000 - 4.0000i
1.0000 + 4.0000i 1.0000 + 4.0000i 3.0000 - 4.0000i 3.0000 - 4.0000i
1.0000 + 4.0000i 1.0000 + 4.0000i 3.0000 - 4.0000i 3.0000 - 4.0000i
Columns 5 through 8
0 - 1.0000i 0 - 1.0000i -1.0000 - 4.0000i -1.0000 - 4.0000i
0 - 1.0000i 0 - 1.0000i -1.0000 - 4.0000i -1.0000 - 4.0000i
0 - 1.0000i 0 - 1.0000i -1.0000 - 4.0000i -1.0000 - 4.0000i
-5.0000 + 3.0000i -5.0000 + 3.0000i 1.0000 - 3.0000i 1.0000 - 3.0000i
-5.0000 + 3.0000i -5.0000 + 3.0000i 1.0000 - 3.0000i 1.0000 - 3.0000i
-5.0000 + 3.0000i -5.0000 + 3.0000i 1.0000 - 3.0000i 1.0000 - 3.0000i
3.0000 - 5.0000i 3.0000 - 5.0000i 3.0000 - 3.0000i 3.0000 - 3.0000i
3.0000 - 5.0000i 3.0000 - 5.0000i 3.0000 - 3.0000i 3.0000 - 3.0000i
3.0000 - 5.0000i 3.0000 - 5.0000i 3.0000 - 3.0000i 3.0000 - 3.0000i
Columns 9 through 10
4.0000 + 1.0000i 4.0000 + 1.0000i
4.0000 + 1.0000i 4.0000 + 1.0000i
4.0000 + 1.0000i 4.0000 + 1.0000i
2.0000 - 2.0000i 2.0000 - 2.0000i
2.0000 - 2.0000i 2.0000 - 2.0000i
2.0000 - 2.0000i 2.0000 - 2.0000i
-3.0000 - 3.0000i -3.0000 - 3.0000i
-3.0000 - 3.0000i -3.0000 - 3.0000i
-3.0000 - 3.0000i -3.0000 - 3.0000i
w=cross(x-xp,u)
w =
-13.0000 +11.0000i
-1.0000 +16.0000i
3.0000 -28.0000i
dot(conj(w),x-xp)
ans =
0
w=conj(cross(x-xp,u))
w =
-13.0000 -11.0000i
-1.0000 -16.0000i
3.0000 +28.0000i
dot(w,x-xp)
ans =
0
dot(w,u)
ans =
0 +1.0658e-014i
diary off