# Tagged Questions

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### Eigenvalue problem: Prove that all of the eigenvalues of $A$ are 1.

Here's a cute problem that was frequently given by the late Herbert Wilf during his talks. Problem: Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. ...
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### Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$

Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues. If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
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### Eigenvalues of product of a matrix and a diagonal matrix

My situation is as follows: I have a symmetric positive semi-definite matrix L (the Laplacian matrix of a graph) and a diagonal matrix S with positive entries $s_i$. There's plenty of literature on ...
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### How to prove Weyl’s asymptotic law for the eigenvalues of the Dirichlet Laplacian?

The following comes from Springer Online Reference Works: Consider a bounded domain $\Omega\subset\mathbb{R}^n$ with a piecewise smooth boundary $\partial\Omega$. $\lambda$ is a Dirichlet eigenvalue ...
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### Matrix Inverses and Eigenvalues

I was working on this problem here below, but seem to not know a precise or clean way to show the proof to this question below. I had about a few ways of doing it, but the statements/operations were ...
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### Why are all nonzero eigenvalues of the skew-symmetric matrices pure imaginary?

Assume that $A$ is an $n\times n$ skew-symmetric real matrix, i.e. $$A^T=-A.$$ Since $\det(A-\lambda I)=\det(A^T-\lambda I)$, $A$ and $A^T$ have the same eigenvalues. On the other hand, $A^T$ and ...
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### Possible eigenvalues of a matrix satisfying certain conditions

This problem appeared in my algebra textbook Suppose $A$ is a real matrix such that $A^2 = A^t.$ What are the possible eigenvalues of $A?$ Here is what I tried in order to solve it. Since the ...
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### similar matrices have same eigenvalue… stuck

So $Ax=\lambda x$ where $A$ is the matrix, $\lambda$ and $x$ its eigenvalue and eigenvector resptectively. Now I have another matrix similar to $A$ i.e. $B=TAT^{-1}$ I'll let a vector $y$ which is ...
Let $A_{n\times n}$ be Hermitian with eigenvalues $\lambda_1 > \lambda_2 > \ldots > \lambda_r=0$ and multiplicities $q_1,...,qr$. Can $A$ be diagonalized? Is the matrix of eigenvalues ...
### Is there a connection between the diagonalization of a matrix $A$ and that of the product $DA$ with a diagonal matrix $D$?
Given a diagonalizable matrix $A = P_0\Lambda_0 P_0^{-1}$ and a diagonal matrix $D$ with $\det D=1$, is there any connection between $P_0$ and the matrix $P$ of the diagonalization of \$DA = P\Lambda ...