# Questions tagged [matrix-congruences]

35 questions
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### Similarity of matrices and its square root over $\mathbb Z$

I already ask this but now its "for all" Prove or disprove: $A \in M(3,\mathbb{Z})$ has a square root with integer entries if and only if $XAX^{-1} \in M(3,\mathbb{Z})$ has a square root with ...
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### Two real symmetric matrices are congruent if and only if they have the same rank and signature.

So I saw this statement in an exercise : Two real $n \times n$ matrices are congruent if and only if they have the same rank and the same signature. But I was wondering why do we need to state the ...
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### Congruent matrices - why do we require invertiblility?

If $A$, $B$ $\in K^{n \times n}$ are $n \times n$ matrices over a field $K$, then we say that $A$ and $B$ are congruent if there exists an invertible $P \in GL(n, K)$ such that $B = P^TAP$, where $P^T$...
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### eigenvalues of a matrix and its product with a diagonal matrix

There is a similar question to mine posted here. I have a matrix $L$ which is the graph Laplacian of a strongly connected normal graph. Therefore, $L$ is normal, has a simple eigenvalue at zero, and ...
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### Finding matrix A in terms of matrix C in W*C=A*W

I am solving an equation and I have a problem. In the right-hand side of my equation; I have everything in the form of "(G+H+T)×W" except one term "W×C". (C,G,H,T are all constant matrices(n*n), and W ...
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### What happens when two diagonal matrices are unitarily similar?

I'm given two diagonal matrices $D$ and $E.$ And I've managed to show that $D=UEU^*$ where $U$ is a unitary matrix. I have to show that $D$ and $E$ are related to a permutation: $D=PEP^T$ where P is ...
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### Does congruence preserve the reality of matrix eigenvalues?

Let $A \in \mathbb{R}^{n\times n}$ have real eigenvalues. Does $$B = X^TAX$$ also have real eigenvalues, for $X \in \mathbb{R}^{n\times n}$ and invertible? Incidentally ...