对称三对角矩阵的特征值分解及相关矩阵求解（MATLAB实现）

Hey there, let's tackle this problem step by step. We need to work with that specific symmetric tridiagonal matrix $A$, compute the required matrices $U\sqrt{\Sigma}$ and $\sqrt{\Sigma}^{-1}UT$ where $U\Sigma^2UT = A$, all using MATLAB. Here's how to do it:

1. First, build the target matrix $A$ #

Looking at the structure of $A$:

• The main diagonal entries follow the pattern $2N-1, 2N-3, ..., 1$ (for row/column $i$, it's $2N - (2i-1)$)
• The off-diagonal entries (both upper and lower) are $N-1, N-2, ..., 1$ (for the $i$-th off-diagonal position, it's $N-i$)

We can construct this easily in MATLAB with the diag function:

function A = build_target_matrix(N)
    % Generate main diagonal elements
    main_diag = 2*N - (1:2:2*N-1);
    % Generate off-diagonal elements
    sub_diag = (N-1):-1:1;
    % Assemble the symmetric tridiagonal matrix
    A = diag(main_diag) + diag(sub_diag, 1) + diag(sub_diag, -1);
end

Test it with $N=3$ and you'll get the expected matrix:
$$
\begin{bmatrix}
5 & 2 & 0 \
2 & 3 & 1 \
0 & 1 & 1
\end{bmatrix}
$$

2. Perform eigenvalue decomposition #

Since $A$ is symmetric, MATLAB's eig function will give us an orthogonal matrix $U$ and a diagonal eigenvalue matrix $\Lambda$ such that $A = U\Lambda U^T$. This is exactly the decomposition we need, because $\Sigma^2 = \Lambda$ (so $\Sigma$ is the square root of $\Lambda$).

% Set your desired N value here
N = 5;
% Build the matrix
A = build_target_matrix(N);
% Get orthogonal U and eigenvalue matrix Lambda
[U, Lambda] = eig(A);
% Compute Sigma as the square root of Lambda (diagonal matrix)
Sigma = sqrt(Lambda);

3. Calculate the target matrices #

Now we just compute $U\sqrt{\Sigma}$ and $\sqrt{\Sigma}^{-1}UT$. Since $\Sigma$ is diagonal, its square root is also diagonal (entries are the square roots of $\Sigma$'s entries, which are the 4th roots of $A$'s eigenvalues). The inverse of $\sqrt{\Sigma}$ is simply the diagonal matrix with entries that are reciprocals of $\sqrt{\Sigma}$'s entries.

% Compute sqrt(Sigma)
sqrt_Sigma = sqrt(Sigma);
% Calculate U*sqrt(Sigma)
U_sqrtSigma = U * sqrt_Sigma;

% Compute inverse of sqrt(Sigma)
inv_sqrtSigma = inv(sqrt_Sigma);
% Calculate sqrt(Sigma)^-1 * U^T
inv_sqrtSigma_Utrans = inv_sqrtSigma * U';

4. Verify your results #

To make sure everything checks out, run these quick verifications:

• Confirm $U$ is orthogonal: norm(U'*U - eye(N)) should be very close to 0 (within floating-point error)
• Confirm $U\Sigma^2UT = A$: norm(U * Sigma^2 * U' - A) should also be near 0
• Check that $(U\sqrt{\Sigma})(\sqrt{\Sigma}U^T)$ squared equals $A$: norm( (U_sqrtSigma * (sqrt_Sigma * U'))^2 - A ) should be negligible

A quick note #

This matrix $A$ is positive definite (strictly diagonally dominant with positive diagonal entries), so all eigenvalues are positive—no issues with complex numbers when taking square roots or inverses.

内容的提问来源于stack exchange，提问作者Olle Kjellqvist