特征值和特征向量的方程组 [英] System of equations for Eigenvalues and Eigenvectors

问题描述

特征值和特征向量可以通过以下等式从张量T导出.

An Eigenvalue and Eigenvector can be derived from the Tensor T by the below equation.

我试图得到一个特征值，特征向量和张量T的方程组，以推导T.

I am trying to get a system of equations for Eigenvalues, Eigenvectors and the Tensor T to derive T.

T矩阵方程为:

(T(i,k)-L(r)*I) * A(r,k) = 0

第一个条目应该是:

[(T11-L1)*A11 T12*A12       T13*A13        T14*A14     ]
[T21*A11      (T22-L1)*A12  T23*A13        T24*A14     ]   
[T31*A11      T32*A12       (T33-L1)*A13   T34*A14     ]
[T41*A11      T42*A12       T43*A13        (T44-L1)*A14]

推荐答案

首先，让我们使用

First, let's declare the symbolics easier using sym:

T = sym('T%d%d', [4 4]);
A = sym('A%d%d', [4 4]);
L = sym('L', [4 1]);

原始代码有几个问题； 1. f在每次内部迭代中都会被替换. 2..内部结果应为标量，因此I不得出现在此处. (请注意，您也可以像 eye(4) 这样定义I手动编写.)

There are several problems with the original code; 1. f is being replaced in each inner iteration. 2. The inner result should be scalar and thus I must not appear there. (Note that you can also define I like eye(4) instead of writing it manually.)

这是更正的版本:

f = cell(4,1); % Initialize equation system

for r = 1:k
    for k = 1:4
        for i = 1:4
            f{r}(i,k) = T(i,k) * A(r,k);
        end
    end
    f{r} = f{r} - L(r)*diag(A(r,:));
end

f{i}将是ith切片.

注意:正如@Schorsch所指出的(Matlab还显示了警告)，总是尝试使用除i(或j)之外的其他变量名. ，因为它们代表 虚构单元 .

Note: As @Schorsch pointed out (and Matlab also shows a warning) always try to use another variable name other than i (or j), since they represent the imaginary unit.

只是为了好玩，您可以使用 repmat 删除两个内部循环:

Just for fun you can use repmat to remove the two inner loops:

for r = 1:4
    f{r} = T .* repmat(A(r,:), [4 1]) - L(r)*diag(A(r,:));
end

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