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Questions tagged [matrix-congruences]

For questions about congruent matrices.

Two matrices $A$ and $B$ over a ring are called congruent if there exists an invertible matrix $P$ over the same ring such that $$P^TAP=B.$$ Matrix congruence is an equivalence relation.

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Did I just discover a new way to calculate the signature of a matrix?

Due to the complains for more clarity down below I've cut my post into segments. Feel free to skip right to Definitions, Algorithm & Conjecture. If this is not clear enough, then I'm afraid I can't help it. Story I'm taking a course on linear…
Figment
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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 $P^T$ denotes the transpose of $P$. Why do we require…
Irregular User
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Matrix congruences

If A, B are two integer square matrices of the same size such that $A\equiv B\pmod n$, is $A^p\equiv B^p \pmod{pn} $ for a prime p dividing n?
Asvin
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If $AA^T$ is a diagonal matrix, what can be said about $A^TA$?

I am trying to answer this question and any method I can think of requires a knowledge of $A^TA$ given that $AA^T=D$, where $D$ is diagonal and $A$ is a square matrix. I could not find anything useful in MSE or elsewhere and I was unable to do any…
Joca Ramiro
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Eigenvalues and eigenvectors after a congruence transformation

Say I have a symmetric matrix $A$ and a symmetric matrix $B$ such that $B$ is congruent with $A$, i.e. there exists a non-singular matrix $X$ such that $B = X^TAX$. Is there a general relation between the eigenvectors and eigenvalues of $A$ and $B$…
lcortesh
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Why is the first $p$-adic congruence subgroup a pro-$p$ group?

I am trying to see that $\Gamma_2$, defined as the kernel of the natural surjective map $\text{GL}_2(\mathbb Z_p)\to \text{GL}(\mathbb F_p)$ is a pro-$p$ group. So I'm trying to show that every finite quotient is a $p$-group. As my first attempt,…
Ariadne
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Is every skew-symmetric matrix congruent to a diagonal matrix?

Question Prove/disprove: if A, a matrix nxn over field F is skew-symmetric then A congruents with a diagonal matrix. My thoughts I know that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. But is it true for…
Splash
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Further explanation wanted on 'double Gaussian elimination' to triangularize a matrix.

I am trying to learn a more efficient way to triangularize a matrix. I found the following answer here on StackExchange which I found interesting, talking about 'double Gaussian elimination': Short way for upper triangularization Would someone be so…
Newbie1000
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Interpretation of *congruence for complex matrices

Recently I've been learning *congruence on complex matrix. The definition of *congruence is that: Let $A, B \in \mathsf{M}_n$. $A$ and $B$ are *-congruent if and only if there exists $S \in \mathsf{M}_n$ such that $A = SBS^\ast$. A few notions come…
nekodesu
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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 all its other eigenvalues have positive real…
WorkInProgress
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Change basis so that a positive definite matrix $A$ is now seen as $I$.

I was solving some physics problems with linear algebra and found this : Denote the basis for the vector space as $\mathbf e_i$, $i,1,...,n$. Consider a change of basis $\mathbf e_i\rightarrow \mathbf e'_i$ in which the components of the velocity…
Vladimir Vargas
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$A ^3$ is congruent to $A$ for a singular symmetric real matrix $A$

I am asked to show that $A ^3$ is congruent to $A$ for all symmetric real matrices $A$. If $A$ is invertible, then - $A^3 = A * A * A = A ^t * (A) * A$ and they are congruent by definition, because $A$ is non-singular and symmetric as stated. I am…
Eliyahu Abadi
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A symmetric matrix that is similar to a diagonal matrix

Perhaps someone here can help me with a homework exercise. Given a symmetric matrix $A$, find an orthogonal matrix $B$ such that $B^tAB=D$ is a diagonal matrix whose entries are arranged in ascending order. Moreover, choose $B$ so that in each…
Andre
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Does congruence transformation preserve definiteness of a nonsymmetric matrix?

Let $A$ be a nonsymmetric negative definite matrix, i.e., $x^\top (A+A^\top) x < 0$. If we invoked a congruent transformation, i.e., $DAD^\top=B$ where $D$ is a nonsingular matrix, will the resulted matrix $B$ still remain negative definite? Can it…
palas
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Matrix congruence - find transition matrix

Let $A,B\in R^{n\times n}$ be symmetric matrices. Given a matrix congruence relation: $$ B = P^TAP $$ Is there an analytical solution or numerical algorithm for finding the transition matrix P?
user415012
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