Questions tagged [matrix-congruences]
For questions about congruent matrices.
66 questions
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The possible number of equivalence classes for the invariants of two congruent matrices
Hi i was doing the exercize 26 section 6.8 of Linear Algebra Friedberg et al., the exercize says: prove that the number of possible equivalence classes between two congruent matrix (so the number of ...
- 55
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25
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PCA analog when points are SPD matrices and congruence instead of linear projection
Let's consider the following formulation of PCA. We have a set of points $\{x_1, ..., x_m\}$ in $\mathbb{R}^n$, and a function $L_{\mathbf{V}}: \mathbb{R}^n \to \mathbb{R}^k$, with $k < n$, where $...
- 152
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Symmetric matrix diagonalization on matrix of eigenvalues via congruence
I have a positive semi-definite symmetric matrix $M\in\mathbb{Q}^{n\times n}$ with Eigenvalues $\lambda_i\in \mathbb{Q}$, $i=1,...,n$ and at most one Eigenvalue equal to $0$.
Is there a way to bring $...
- 1,355
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2
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68
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Congruent diagonalization using row and column operations
Let $$A=\begin{pmatrix}
1 & 2 & 3\\
2 & 4 & 6\\
3 & 6 & 9 \end{pmatrix}.$$
Find an invertible matrix $P$ such that $P^tAP$ is diagonal.
Let me start by saying that I already ...
- 1,392
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40
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Find values which make two matrices congruent
For which values of $a\in\mathbb{R}$ are the following matrices congruent?
$$A=\begin{pmatrix} 1&4-a-a^2\\
2& -1 \end{pmatrix}$$
$$B=\begin{pmatrix} -a-1 & 3\\
3 & -5
\end{pmatrix}$$
...
- 1,392
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0
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46
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Find diagonal matrix $D$ such that $A$ is congruent to $D$
Let $A\in M_n(\mathbb{R})$ be the matrix with all entries equal $1$. Find the signature of $A$ and find a diagonal matrix $D$ such that $A$ is congruent to $D$.
The characteristic polynomial of $A$ ...
- 1,392
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Congruency of two matrices over the field of rational numbers.
Consider the matrix $A=\begin{bmatrix}0&1\\
1&0
\end{bmatrix}$ over the field $\Bbb Q$ of rationals.Which of the following matrices are congruent to $A$ over $\Bbb Q$?
$(1).$ $A_1=\begin{...
- 646
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1
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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 ...
- 351
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52
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Determinant of $ADA^\top$, where $D$ is diagonal and $A$ is wide
Let $A\in\mathbb{R}^{M\times N}$ be full rank, with $M<N$, and let $D\in\mathbb{R}^{N\times N}$ be diagonal, with a strictly positive diagonal.
Is it possible to simplify $\det{ADA^\top}$? In ...
- 667
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138
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How can I tell if these matrices are congruent?
I am completey lost on this. I know a matrix $B$ is congruent to $A$ if $B = P^\top\!\!AP$ but I tried finding the e-vectors and e-values for $P$ and $P^\top\!$ to spit out $A$ again. I really don't ...
- 39
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192
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Some clarification over matrix congruence, matrix similarity, and the (finite-dim) spectral theorem
In linear algebra, we have the following well-known result.
Proposition. Every real symmetric matrix $A$ is congruent to a diagonal matrix with real eigenvalues on the diagonal. That is, $A=P^T \...
- 2,020
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106
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Canonical form of a hermitian matrix with respect to congruence
Just a little warning: my english is not that good so the text below can be really confusing.
How can I proof that every hermitian matrix, $H\in C^{n\times n}$, is congruent to a matrix:
\begin{align}
...
- 11
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50
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telling if two matrices are congruent and computing the N matric s.t A=N$^T$BN
i guess that to matrices are congruent if and only if they have the same signature, but then if i know that they are congruent, how can i find the N matrix?
for example having $A = \begin{bmatrix}
10 &...
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65
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Congruence between Symmetric and non-Symmetric Matrices for Quadratic Form
I learned that for a bilinear form/square form the following theorem holds:
Matrices $A,B$ are congruent if and only if $A,B$ represent the same bilinear/quadratic form.
Now, suppose I have the ...
- 1,729
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69
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Could we tell whether two matrices are congruent only by its eigenvalues? if not, How many conditions do we need?
When I was learning in my junior high, I am fascinated by the fact that There are more than one way like SSS, SAS etc to tell whether two triangles are congruent or not. Thus I am wondering, Is there ...
- 5
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$A,B \in M_3(\mathbb{R}) $ such that $A,B$ are symmetric. Find all values of $k\in \mathbb{N}$ such that $A^k , B^k$ are congruent.
$A,B \in M_3(\mathbb{R}) $ such that $A,B$ are symmetric.
$P_A(x)=x^3-x$ (the characteristic polynomial of $A$).
$P_B(y)=y^3-3y^2+2y$ (the characteristic polynomial of $B$).
Find all values of $k\in \...
- 2,322
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Congruence of Matrices in Knot theory
I've been learning about knot theory lately and got stuck on a proof in Cromwells 'Knots and links', where some linear algebra is needed. For anyone interested its on pg. 158.
In the proof he wants to ...
- 1
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2
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397
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How to convert this symmetric matrix to normal form? [closed]
I have to find the signature of the symmetric matrix $\begin{pmatrix}1 & 0 & 0 &0\\0 &0 &1 &0\\0&1&0 &0\\0& 0 &0 & 1\end{pmatrix}$. Now to find the ...
- 3,801
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Calculate maximum number of independent columns of a modular matrix
I have a question on determining the minimum number of dependent columns (of the maximum number of independent ones) of a matrix when the coefficients are all modulo $m\in \mathbb{N}$.
For example, ...
- 2,096
5
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111
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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?
- 8,369
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Is Same signs of determinants of $A$ and $B$ $\implies P^tAP =B$
Let $$A =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & b \\
\end{pmatrix}
$$...
- 523
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0
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80
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Show the relation between singular values of $A$ and $B^*AB$
Proposition: For any nonsingular $A\in\mathbb{C^{n\times n}}$ and nonsingular $B\in\mathbb{C^{n\times n}}$, the following holds
$$\frac{\Re(x_i^*Ax_i)}{\sigma_i(A)}=\frac{\Re(y_j^*B^*ABy_j)}{\sigma_j(...
- 1,930
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1
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425
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How to conclude that matrices $A$ and $B$ are congruent?
Let $A$ and $B$ be real $n×n$ matrices. Which of the following statements is false ?
If $A^{-1}$ and $B^{-1}$ are congruent then so are $A$ and $B$.
If $A^t$ and $B^t$ are congruent then so are $A$ ...
- 523
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54
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Showing that a certain binary matrix cannot be congruent to the null matrix
I don't get why the following matrix (whose entries belong to the binary field)
\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
cannot be congruent to the null matrix ,according to my notes.
Can ...
- 121
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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'...
- 257
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Congruently and Hermitely diagonalisable matrices
It is known that a a matrix is diagonalisable (by the similarity equivalence relation) if and only if there exists a basis of eigenvectors. A typical course in linear algebra then gives two additional ...
- 113
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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$...
- 177
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2
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Show that a matrix A above the R field is congruent to itself squared iff A is non-negative
How do I show that a symmetric matrix A above the R field is congruent to itself squared iff A is non-negative, but I'm not sure if by non negative the author meant positive semi-definite:
$$\exists P:...
- 1,298
2
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2
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680
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A symmetric matrix that is similar to a diagonal matrix [closed]
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 ...
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207
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How to find the normal form under congruence of the given matrix.
Consider the matrix $$\begin{bmatrix}
1&-1&-1\\
-1&1&-1\\
-1&-1&1\\
\end{bmatrix}$$.
Suppose we want ...
- 3,801
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2
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2k
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How to reduce a quadric into canonical form without rigorous calculations?
Consider the quadric in $(x,y,z)$ given by $x^2+y^2+z^2-2xy-2yz+2zx+x-4y+z+1=0$.I am asked to reduce it into canonical form and describe the nature.Here it is a parabolic cylinder.But the method ...
- 3,801
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112
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Kronecker-Weierstrass problem, 3x6 matrices conjugacy or congruent classes?
I'm here again with a somewhat vague and hard question our teacher asked us, we have to check and proof that for all matrices $A$ and $B$ that by applying certain simultaneous transformations, we can ...
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1
answer
66
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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 \...
- 185
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0
answers
44
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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 $\operatorname{M}(3,\mathbb{Z})$....
- 185
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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 ...
- 185
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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 $...
- 185
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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+...
- 11
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0
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819
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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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1
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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$.
(b) $B=SAS^...
- 1,381
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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 ...
- 1,619
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80
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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 ...
- 257
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1
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687
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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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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?$$
Does this matrix necessarily ...
- 257
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0
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70
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How to show if a non-(skew) symmetric matrix is congruent/orthogonal equivalent to a diagonal matrix
I'm currently preparing for an exam, and I'm getting stuck at this seemingly straightforward question. The question is to check if the matrix $A$, given below, is congruent and/or orthogonal ...
- 497
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1
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203
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Check if there exists an orthogonal matrix $P$ such that $B=P^{-1}AP$
I want to check if for matrices
$$
A =
\begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 ...
- 2,539
1
vote
0
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1k
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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 ...
- 101
1
vote
0
answers
253
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Congruence 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.
2
votes
0
answers
330
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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?
- 21
4
votes
1
answer
439
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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 ...
- 128
1
vote
0
answers
1k
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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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