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Questions tagged [matrix-congruences]

For questions about congruent matrices.

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Square root of a specific matrix over $\Bbb Z$

Let $ B^2 = \begin{bmatrix} -2&0&0 \\ -1&-4&-1\\ 2&4&0\\ \end{bmatrix}^2 = \begin{bmatrix} 4&0&0\\ 4&12&4\\ -8&-16&-4\\ \end{bmatrix} = A similar to \...
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Square root and similarity between integer matrices

Prove or disprove: Let $A, B \in \operatorname{M}(3,\mathbb{Z})$ and $A \sim B$. $A$ has a square root in $\operatorname{M}(3,\mathbb{Z})$ iff $B$ has a square root in $\operatorname{M}(3,\...
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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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43 views

Square root of a matrix $A$ and matrices similar to $A$

Prove or disprove: $A \in \Bbb M(3,\mathbb{Z})$ has a square root with integer entries if and only if $XAX^{-1} \in \Bbb M(3,\mathbb{Z})$ has a square root with integer entries, for some invertible $...
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How to solve system of linear congruences with the same modulo?

I have to write program which is solving linear congruences withe the same modulo. I have system of congruences like that(only 2 unknowns x and y): $$\begin{cases} a_1x+b_1y \equiv c_1\pmod n \\ a_2x+...
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242 views

Show that congruence of matrices is an equivalence relation.

How would one solve a question of this nature? We know that given an arbitrary square matrix, A, that a matrix B is said to be congruent to A if there exists a nonsingular (invertible) matrix P such ...
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1answer
212 views

Number of distinct equivalence classes under *-congruence and T-congruence

Let me first state the definition Let $A,B\in\mathbb{C}^{n\times n}$. If there exists a nonsingular matrix $S$ s.t. (a) $B=SAS^*$, then $B$ is said to be *-congruent or conjuctive to $A$. ...
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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 ...
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Proof that any antisymmetric matrix C is congruent to a block diagonal matrix?

Is there a simple proof that shows that, for any antisymmetric $C$, there exists an orthogonal matrix $P$ such that $P^TCP$ is a block diagonal matrix? I have found a couple of very longwinded ...
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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 ...
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1answer
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Congruence transformation of symmetric matrices [closed]

Given a symmetric matrix $A$ of size $n$ and an arbitrary invertible square matrix $P$ (also of size $n$), what can we say about the congruence transformation: $$P^TAP?$$ Does this matrix necessarily ...
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How to show if a non-(skew) symmetric matrix is congruent/orthogonal equivalent to a diagonal matrix

I'm currently preparing for an exam, and I'm getting stuck at this seemingly straightforward question. The question is to check if the matrix $A$, given below, is congruent and/or orthogonal ...
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1answer
67 views

Check if there exists an orthogonal matrix $P$ such that $B=P^{-1}AP$

I want to check if for matrices $$ A = \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 ...
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What are the properties of congruent classes over symmetric matrices?

Find a property of symmetric bilinear forms on finite-dimensional vector spaces above the stated body of the matrices, which keeps the congruent classes of the given symmetric matrices apart from ...
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Congruency of Skew symmetric matrices

Prove that every matrix congruent to a skew symmetric matrix is skew symmetric. My work- How do I prove that the transforming matrix P is always orthogonal? Please help.
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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?
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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 ...
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Congruence Classes of Quadratic Forms

Let A $\in$ K$^n$$^,$$^n$ be a symmetric matrix over a field K. Define f(A) = { t$^2$ det(A) : t $\in$ K } and g(A) = {$x^T$A$x$ : $x \in $ K$^n$ } . I have shown that if A,B are congruent, then f(A) ...
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Matrix Congruence and Quadratic Forms [closed]

Let A $\in$ K$^n$$^,$$^n$ be a symmetric matrix over a field K. Define f(A) = { t$^2$ det(A) : t $\in$ K } and g(A) = {$x^T$A$x$ : $x \in $ K$^n$ } . WTS if A,B are congruent f(A) = f(B) and g(A) = ...
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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}...
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1answer
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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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1answer
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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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1answer
263 views

Does congruence preserve the reality of matrix eigenvalues?

Let $A \in \mathbb{R}^{n\times n}$ have real eigenvalues. Does \begin{equation} B = X^TAX \end{equation} also have real eigenvalues, for $X \in \mathbb{R}^{n\times n}$ and invertible? Incidentally ...
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prove that a antisymmetric and invertible matrix is congruent to another matrix

let A be an antisymmetric and invertible matrix $A \in M_{2m}(\Bbb{R})$ prove that A is congruent to $$ \begin{pmatrix} 0 & I_m \\ -I_m & 0 \\ \end{pmatrix} ...
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Question on matrices

Let A be a symmetric matrix of form 2.2 then what should be the elements in matrix B of2.2 such that AB not equals to BA
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Discrete version of Sylvester's Law of Inertia

Given a matrix $A\in\mathbb{C}^{n\times n}$, we denote by symbol $\mathrm{In}_d (A):= (n_<,n_>,n_1)$, the discrete inertia (or inertia w.r.t. the unit circle) of $A$, where $n_<,n_>$ and $...
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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 ...
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Is the inverse of a M-Matrix again an M-Matrix?

Is the inverse of a M-Matrix again an M-Matrix?
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What is the Lie group that leaves this matrix invariant?

What is the group that leaves \begin{equation} Y = \bigoplus_{j=1}^{k} \alpha_j \mathbb{I}_{2n_j\times 2n_j} \end{equation} invariant under congruence ($Y = XYX^T$) where $\alpha_j \in \mathbb{R}$ ...
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1answer
679 views

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 \...
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Diagonalizing a bilinear form

Question a. we mark $\mathbb{R}_2[x]$ as the polynomial space of degree $ \le 2$ over the real field $\mathbb{R}$. $\xi :\mathbb{R}_2[x] \times \mathbb{R}_2[x] \to \mathbb{R}$ $$\xi(q,p) = q(-1)p(-...
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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 ...