b = [ 3 ; 2 ; 9]
b =
3
2
9
Big = [A A']
Big =
1 2 3 1 0 7
0 5 6 2 5 8
7 8 9 3 6 9
size(B)
{Undefined function or variable 'B'.}
size(Big)
ans =
3 6
row2part=Big(2,2:4)
row2part =
5 6 2
% take out entire row
row3=Big(3,:)
row3 =
7 8 9 3 6 9
row3=Big(3,1:6)
row3 =
7 8 9 3 6 9
row3=Big(3,2:6)
row3 =
8 9 3 6 9
row3=Big(3,2:end)
row3 =
8 9 3 6 9
Big
Big =
1 2 3 1 0 7
0 5 6 2 5 8
7 8 9 3 6 9
col4=Big(:,4)
col4 =
1
2
3
col345=Big(:,3:5)
col345 =
3 1 0
6 2 5
9 3 6
% delete column 4
AT=A'
AT =
1 0 7
2 5 8
3 6 9
AT(:,2)=[]
AT =
1 7
2 8
3 9
% special matrices
Z=zeros(3)
Z =
0 0 0
0 0 0
0 0 0
Big
Big =
1 2 3 1 0 7
0 5 6 2 5 8
7 8 9 3 6 9
Big+At
{Undefined function or variable 'At'.}
Big+AT
{Matrix dimensions must agree.}
Z=zeros(3,6)
Z =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Z=zeros(size(B))
{Undefined function or variable 'B'.}
Z=zeros(size(Big))
Z =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
size(Big)
ans =
3 6
Big
Big =
1 2 3 1 0 7
0 5 6 2 5 8
7 8 9 3 6 9
Big+Z
ans =
1 2 3 1 0 7
0 5 6 2 5 8
7 8 9 3 6 9
Big*Z'
ans =
0 0 0
0 0 0
0 0 0
I=eye(6)
I =
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
Big
Big =
1 2 3 1 0 7
0 5 6 2 5 8
7 8 9 3 6 9
Big*I
ans =
1 2 3 1 0 7
0 5 6 2 5 8
7 8 9 3 6 9
I=eye(3)
I =
1 0 0
0 1 0
0 0 1
I*Big
ans =
1 2 3 1 0 7
0 5 6 2 5 8
7 8 9 3 6 9
% a matrix A is symmetric if A' = A
S = [3 4 5;
4 6 7;
5 7 8]
S =
3 4 5
4 6 7
5 7 8
S'
ans =
3 4 5
4 6 7
5 7 8
% Eigenvalues and eigenvectors
% If for a squre matrix A.
% A v = lambda v with v not zero
% then v is an eigenvector of A and
% lambda is the corresponding eigenvalue.
C=1
C =
1
S = [ 0 C 0;
C 0 0;
0 0 0]
S =
0 1 0
1 0 0
0 0 0
lambda=eig(S)
lambda =
-1
0
1
lookfor eigenvalue
eigshow - Graphical demonstration of eigenvalues and singular values.
expmdemo3 - Matrix exponential via eigenvalues and eigenvectors.
mat4bvp - Find the fourth eigenvalue of the Mathieu's equation.
rosser - Classic symmetric eigenvalue test problem.
wilkinson - Wilkinson's eigenvalue test matrix.
balance - Diagonal scaling to improve eigenvalue accuracy.
condeig - Condition number with respect to eigenvalues.
eig - Eigenvalues and eigenvectors.
ordeig - Eigenvalues of quasitriangular matrices.
ordqz - Reorder eigenvalues in QZ factorization.
ordschur - Reorder eigenvalues in Schur factorization.
polyeig - Polynomial eigenvalue problem.
qz - QZ factorization for generalized eigenvalues.
eigs - Find a few eigenvalues and eigenvectors of a matrix
dsort - Sort complex discrete eigenvalues in descending order.
esort - Sort complex continuous eigenvalues in descending order.
lyap2 - Lyapunov equation solution using eigenvalue decomposition.
lambda
lambda =
-1
0
1
lambda1=lambda(1)
lambda1 =
-1
lambda2=lambda(2)
lambda2 =
0
lambda3=lambda(3)
lambda3 =
1
%lambda=eig(S)
[E Lambda]=eig(A)
E =
-0.2401 -0.3201 0.3022
-0.4801 -0.6402 -0.7932
-0.8437 0.6983 0.5288
Lambda =
15.5440 0 0
0 -1.5440 0
0 0 1.0000
[E Lambda]=eig(S)
E =
-0.7071 0 0.7071
0.7071 0 0.7071
0 1.0000 0
Lambda =
-1 0 0
0 0 0
0 0 1
e1=E(:,1)
e1 =
-0.7071
0.7071
0
e2=E(:,2)
e2 =
0
0
1
e3=E(:,3)
e3 =
0.7071
0.7071
0
S*e1
ans =
0.7071
-0.7071
0
lambda1*e1
ans =
0.7071
-0.7071
0
lambda1*e1 - S*e1
ans =
0
0
0
lambda2*e2 - S*e2
ans =
0
0
0
lambda3*e3 - S*e3
ans =
0
0
0
S
S =
0 1 0
1 0 0
0 0 0
% the eigenvalues of a (real) symmetric matrix are real
% the eigenvectors of real symmetric matrices can always be taken to be
% of length 1 and mutually orthogonal
v1
{Undefined function or variable 'v1'.}
e1
e1 =
-0.7071
0.7071
0
e2
e2 =
0
0
1
e1'*e2
ans =
0
e1*e3
{Error using *
Inner matrix dimensions must agree.}
e1'*e3
ans =
0
e2'*e3
ans =
0
e1'*e2
ans =
0
e1'*e1
ans =
1.0000
e2'*e2
ans =
1
e3'*e3
ans =
1.0000
E
E =
-0.7071 0 0.7071
0.7071 0 0.7071
0 1.0000 0
E'*E
ans =
1.0000 0 -0.0000
0 1.0000 0
-0.0000 0 1.0000
% For complex matrices, the same observation only