# Questions tagged [diagonalization]

For questions about matrix diagonalization. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. This tag is NOT for diagonalization arguments common to logic and set theory.

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### block diagonalizing a 4x4 Hermitian matrix with vanishing anti diagonal elements

I need help diagonalizing the following 4x4 matrix that has a vanishing anti-diagonal: \begin{align} H = \begin{pmatrix} a & b & c & 0 \\ b^* & d & 0 & e \\ c & 0 & f &...
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### How to show a matrix DAD has distinct eigenvalues, where D is a diagonal matrix and A is a highly structured matrix

If D is a positive diagonal matrix with well-separated diagonal entries (in particular, $(1 + k) |D_{i - 1, i - 1} < D_{i, i} < (1 - k) D_{i + 1, i + 1}$, where $k$ is a constant and the ...
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### Eigenvectors of two commuting diagonalizable matrices when the eigenspaces need not have dimension one

Let $A,B$ be commuting diagonalizable $n\times n$ matrices over $\Bbb C$. Suppose that the eigenvalues of $A$'s are all distinct (so the eigenspaces have dimension one), and the same for $B$. Then any ...
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### Why is there not a test for diagonalizability of a matrix

Let $A$ be square.This question is a bit opinion based, unless there is a technical answer.I think it is helpful tho. Also this question is closely related to this question : quick way to check if a ...
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### Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices

Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2 ,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume ...
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### Verify that a quadratic form is NOT positive definite

Verify that the quadratic form $$q(x_1,x_2,x_3)=x_1^2+4x_1x_2+3x_2^2+2x_2x_3+6x_3^2$$ is NOT positive definite and find a vector in $v\in\mathbb{R}^3$ such that $q(v)<0$ . I have made several ...
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