最基本的特徵值問題分為三類：

1、標準的線性特徵值問題：

Ax=λx,A∈Cn∗nAx=λx,A∈Cn∗n

2、普遍的線性特徵值問題：

Ax=λBx,A、B∈Cn∗nAx=λBx,A、B∈Cn∗n

3、普遍的艾米特正定線性特徵值問題：

Ax=λBx,A、B∈Cn∗nAx=λBx,A、B∈Cn∗n

A∗=A,B∗=B>0∈Cn∗nA∗=A,B∗=B>0∈Cn∗n

Arnoldi方法求特徵值是特別常用的一個方法，matlab的內建函式eigs就是用了這個方法，一起來看看。

% demo routine for the usage of MATLAB eigs function.

% Example: Compute eigenvalues of a matrix A close to sigma,

% using shift and invert spectral transformation.

rng(0);%設定隨機種子

A = RandomWalkProb(15);%產生隨機矩陣

n = length(A);%矩陣長度

sigma = 0.01; %最終是為了找出離sigma最近的特徵值

[L, U, P] = lu(A - sigma * eye(n));%LU分解, L*U = P*A,此因A不一定能LU分解

Afun = @(x) U \ (L \ (P*x));%Afun其實就是1/(A-sigmaI)

opts.isreal = true;%1 if A or A-sigma*B represented by Afun is real, 0 otherwise.

opts.issym = false;%1 if A or A-sigma*B represented by Afun is symmetric, 0 otherwise.

opts.disp = 2;%Diagnostic information display level.

num = 3;

[V, E] = eigs(Afun, n, num, sigma, opts);%呼叫了eigs函式，對求距離sigma最近的num個矩陣

%結果最後V是特徵向量組成的矩陣，E的對角線是特徵值

% END

其中的RandomWalkProb.m:

function A = RandomWalkProb(k)

% Function A = RandomWalkProb(k) generate a stochastic matrix A

% of size n = (k+1)*(k+2)/2.

%

% References:

% Y. Saad, Numerical Methods for Large Eigenvalue Problems, Revised ed.

% SIAM, Philadelphia, 2011. pp 36--37.

%

II = []; JJ = []; VV = [];

idx = @(i,j) i*(2*k-i+3)/2+j+1;

for i = 0:k

for j = 0:k-i

% (i,j) --> (i-1, j)

if i>0

II = [II, idx(i,j)];

JJ = [JJ, idx(i-1,j)];

VV = [VV, (i+j)/2/k * (1+(j==0))];

end

% (i,j) --> (i, j-1)

if j>0

II = [II, idx(i,j)];

JJ = [JJ, idx(i,j-1)];

VV = [VV, (i+j)/2/k * (1+(i==0))];

end

% (i,j) --> (i+1, j)

if j+i<k

II = [II, idx(i,j)];

JJ = [JJ, idx(i+1,j)];

VV = [VV, 0.5 - (i+j)/2/k];

end

% (i,j) --> (i, j+1)

if j+i<k

II = [II, idx(i,j)];

JJ = [JJ, idx(i,j+1)];

VV = [VV, 0.5 - (i+j)/2/k];

end

end

end

A = sparse(II,JJ,VV);

return;