Questions tagged [symmetric-matrices]
A symmetric matrix is a square matrix that is equal to its transpose.
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16 views
For what kind of matrix $A$, there is a (symmetric) positive definite matrix $B$ such that $BA$ is symmetric
Let $A$ be a $n\times n$ real matrix. My ultimate goal is to find a sufficient condition on $A$ such that all the eigenvalues of $A$ are real.
Therefore, I want $A$ to be self-adjoint with respect ...
1
vote
0answers
13 views
Geometric series of binary relation
Let $\rho\subset X\times X$ be a symmetric binary relation on a finite set $X$. Let $\overline{\rho}\subset X\times X$ be its transitive closure :
$$
\overline{\rho}=\bigcup_{i=0}^\infty \rho^{\circ i}...
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0answers
19 views
Efficiently finding a single row of the inverse of a symmetric (not Hermitian) complex matrix
I want to find, at many frequencies $w$, the response at a few nodes to input at one node in a 1-D kinematic system with constraints, i.e. find (part of) $\mathbf x$ s.t. $$\left(- w^2 \mathbf M + jw \...
1
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0answers
7 views
Are there any interesting properties of the following symmetric circulate matrix?
Consider $a_0, \ldots, a_{k-1} \in \mathbb{R}$, consider matrix $\mathbf{A}$ as the following
$$\mathbf{A} = \begin{bmatrix} a_{0} & a_{1} & \ldots & a_{k-1} \\
a_{1} & a_{0} & \...
2
votes
2answers
41 views
Prove eigenvalues of a symmetric matrix are in a certain interval
I am given a matrix $A=\begin{bmatrix}1&2&0\\0&1&2\\0&0&1\end{bmatrix}$. I am asked to compute $A^tA=\begin{bmatrix}1&2&0\\2&5&2\\0&2&5\end{bmatrix}$
...
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1answer
14 views
For a positive semi-definite $d\times d$ matrix $A$, $ (\text{det}(AS))^{\frac{1}{d}}\leq\frac{1}{d}\text{Tr}(AS) $ for every $S\in\text{SPD}_{d}$.
For a positive semi-definite $d\times d$ matrix $A$, $$
(\text{det}(AS))^{\frac{1}{d}}\leq\frac{1}{d}\text{Tr}(AS) $$ for
every $S\in\text{SPD}_{d}$.
I would like to show the above statement. If ...
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vote
2answers
40 views
Diagonalization of alternating matrix
I am reading the book "Algebra", written by Serge Lang and having difficulty in an explaining from that book on page 588.
The problem is the following.
Let $G \in M_n(\mathbb{R})$ be an alternating ...
0
votes
1answer
44 views
Trace inequality for a symmetric matrix
Let S be a symmetric positive n×n matrix and $$B \in M_n(R)$$ a triangularizable matrix with spectrum in $$[0, 1]$$
Prove the inequality:
$$tr(BS)\ge tr(B)\det(S)$$
2
votes
3answers
53 views
Inverse of tridiagonal Toeplitz matrix
Consider the following tridiagonal Toeplitz matrix. Let $n$ be even.
$${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}}
{0}&{1}&{}&{}&{}\\
{1}&{0}&{1}&{}&{}\\
{}&...
1
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1answer
32 views
Find a real symmetric matrix $A$
Is there a real symmetric matrix $A$ satisfying the following two conditions?
$A$ is not orthogonal.
There is a positive integer $m >1$, such that
$$A^m=I.$$
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0answers
23 views
I know symmetric matrix $S = QDQ^T$, but how can matrices with form ADA be symmetric?
I have learned that a symmetric matrix must be able to be written in form of $S=QDQ^T$ where Q is the orthonormal eigenvectors. But I saw an example that display a symmetric matrix in the form of $S = ...
2
votes
1answer
117 views
strongly connected $L$, then what are the eigenvalues of $L+L^T$?
I asked a similar question before. Now things become a bit different here.
Suppose $L$ is a non-symmetric Laplacian matrix, of which the corresponding graph is strongly connected. Is it true that $L+...
2
votes
1answer
57 views
Is it true that $L+L^T$ has at most one negative eigenvalue?
Suppose $L$ is a non-symmetric Laplacian matrix, then $L+L^T$ is symmetric. Is it true that $L+L^T$ has at most one negative eigenvalue? I try many examples using Matlab numerically to find out that ...
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0answers
10 views
Minimize double dot product subject to positive definite constraints
Given a symmetric matrix $\varepsilon$ and a 4th order tensor $C$, how can we find the matrix $\delta$ that minimizes $C{:}(\varepsilon{+}\delta){:}(\varepsilon{+}\delta)$ subject to the constraints ...
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2answers
59 views
Proof of orthogonal and symmetric.
Given $x$ is an $n$ dimensional vector, if $A = I_n- (2/x^Tx)xx^T$, show that it is orthogonal and symmetric.
I know that if $A$ is orthogonal and symmetric, $A = \operatorname{inverse}(A) = A^T$, ...
1
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1answer
44 views
Eigenvectors with same eigenvalue
For a symmetric matrix $A$, I am aware that eigenvectors $v_1, \dots, v_n$ with the same eigenvalue $\lambda$ are linearly independent but not orthogonal.
The spectral theorem states that any $p \...
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0answers
16 views
can we perform SVD in spectral clustering to get top-$k$ eigenvector?
In spectral clustering, we need to compute top-k eigenvector for k-means clustering
I have been told that SVD and eigendecomposition is equal for symmetric matrix here page 9.
I try in matlab
<...
2
votes
0answers
30 views
Relationships between top-$k$ eigenvector and top-$k$ singular vector of symmetric matrix $A$
Is there any relationships of top-$k$ eigenvector and singular vector of symmetric matrix $A \in R^{n \times n}$?
For symmetric matrix $A$
its eigenvalue decomposition is:
$$ A = B \Lambda B^T$$
...
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votes
1answer
20 views
Generalizing in matrix form
I have the following expression:
$\sum_{0 < s \leq S} \sum_{0 < a \leq A} (x_{a,s} - \frac{1}{A}\sum_{0 < i \leq A} x_{i,s})^2$
, which I would like to express in matrix quadratic form, as $...
0
votes
1answer
15 views
$LDL^\top$ for symmetric positive semidefinite matrices that are not positive definite
I have a symmetric positive semidefinite matrix (which is not positive definite) with integer entries and I know that I have an $LDL^\top$ decomposition for it (well mainly because Maple gives me one)....
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0answers
42 views
Numerically determine eigenvalues of real non-symmetric matrix known to have positive eigenvalues
I need to numerically determine eigenvalues of real non-symmetric matrix $M$ known to have real positive eigenvalues $\lambda_i>0$. I know this due to the overall problem I'm solving.
Problem is ...
2
votes
1answer
48 views
Eigenvalue properties of matrix whose element-wise absolute value is row-stochastic
Suppose I have a matrix $A\in\mathbb{R}^{n\times n}$ such that the matrix $B=(|a_{ij}|)$ (matrix of absolute values of $A$) is row-stochastic. Suppose also that the eigenvalue $1$ of $B$ is simple. ...
2
votes
2answers
176 views
Find the minimum value of the quadratic form $x^TMx$ and the corresponding $x$, where $M$ is a symmetric matrix
How do you find the minimum value of the quadratic form $x^TMx$ and the corresponding $x$, where $M$ is a symmetric matrix? I have $x^TMx = x^T(I − 2(vv^T)/\|v\|^2)x$, and I don't know how to continue....
2
votes
3answers
86 views
If v ∈ R^n is a nonzero vector, and I ∈ R^n×n is an identify matrix. Prove that M = I − 2(v(v^T)/||v||^2 is symmetric and satisfies M^−1 = M
I thought about showing M = M^T, so M is symmetric. But I don't know how to compute 2(v(v^T)/||v||^2 to find M and M^T. Any idea?
1
vote
1answer
24 views
Conic sections and polynomials over C
Let K be a field with $char(K) \neq 2$. Every polynomial $f \in K[X,Y] $ has a unique representation
$$ a_{11}x^2+a_{22}y^2+2a_{12}xy+ 2a_{13}x + 2a_{23}y + a_{33}$$
that can be identified by a ...
0
votes
1answer
64 views
Matrix equation of the form $C A C^\intercal = D$
Consider the following square matrix
\begin{align}
A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & d & -d+1 & -\frac12 & 0 & ...
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0answers
36 views
Find nth power of symmetric matrix of size m*m where m<=1000 and n can be any positive integer?
i have to find nth power of matrix which is symmetric about main diagonal and all elements of main diagonals are zero.
For example sample matrix
$\left( \begin{array}{cc}
0 & 1 & 0 & 0\\
1 ...
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votes
0answers
16 views
Show inequality for Euclidean norm on SPD matrix and identity matrix
Let A be a symmetric, positive definite matrix. Show that in the Euclidean norm $$||I-\frac{1}{\tau}A||_{2}<1$$ implies that $0<\tau<2||A||^{-1}_{2}$ for $\tau$ a scalar.
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votes
2answers
223 views
What is the limit of $\mathrm{Tr}(G^kM{G^*}^k)^{1/2k}$ when $k$ goes to infinity?
If $G\in \mathscr M_n(\mathbf C)$ then it's well known that $\lim_{k\to \infty}\|G^k\|^{1/k}=\rho(G)$ where $\rho(G)$ is the spectral radius of $G$, the value of the limit does not depend on the ...
1
vote
1answer
48 views
Eigenvalues of a special symmetric matrix
Can somebody help me in finding eigenvalues of the symmetric matrix $ \pmatrix{A & B\\ B & C}$?
Here $A$ and $C$ are symmetric matrices of order $n$ and $B$ is a diagonal matrix of order $n$....
0
votes
1answer
24 views
Maximum Eigenvalue of a Symmetric Matrix!!
Let $M=A-B$ be a symmetric matrix of order n.
I know $\lambda_{max}=\sup_{x\neq 0} \frac{x^tMx}{x^t x}$. Where $ x\in R^n$.
Can I write it like $\lambda_{max}=\sup_{x\neq 0} \frac{x^tAx}{x^t x}-\...
3
votes
0answers
42 views
Find rank of the special block symmetric and persymmetric matrix
I meet a difficult problem recently. The problem is to find the rank of a special matrix:
$$X = \begin{bmatrix} S & P \\ P & JSJ\end{bmatrix}\in \mathbb{R}^{2m\times2m} ,$$ where $S$ is ...
0
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0answers
41 views
When does a product of two symmetric matrices commute?
I have two $n\times n$ symmetric matrices.
I know that $\operatorname{tr}(AB)=\operatorname{tr}(BA)$. But it does not mean $AB=BA$.
I wonder what is the condition for $AB=BA$.
1
vote
1answer
15 views
difference in determinants of Positive Definite Matrices
Let $A$ and $B$ be positive definite matrices (psd) of the same size, such that $A>B$ (i.e. $A-B$ is also psd).
I wonder if $det(A)>det(B)$?
I have tried to find a counter example, but couldn'...
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votes
1answer
30 views
Prove that matrix $\mathbf{A^H A}$ is Hermitian
I have a $M \times N$ matrix $\mathbf{A}$, such that $\mathbf{A^HA}$ is a Hermitian matrix and $M < N$. Is there any way by which I can prove mathematically that $\mathbf{A^H A}$ is Hermitian.
Or ...
1
vote
3answers
61 views
Proof for why symmetric matrices are only orthogonally diagonalizable
I am wondering why symmetric matrices are diagonalizable only by orthogonal matrices (and these orthogonal matrices by definition have orthonormal vectors). This is the proof but I don't really get ...
1
vote
2answers
11 views
“Perturbed projection” matrix equality
I recently came across a throwaway comment in a paper noting the following equality, for an $n\times p$ matrix, $X$, and $\lambda > 0$:
$$I_n - X(X'X + \lambda I_p)^{-1}X' = \lambda(X X' + \lambda ...
2
votes
2answers
73 views
Showing that every rational eigenvalue of a graph is integral
(This is taken from the exercises in Bondy and Murty's Graph Theory.) Let the adjacency matrix of a graph $G$ be denoted by $\mathbf{A}$. The eigenvalues $\lambda$ of $G$ are defined as the roots of ...
3
votes
2answers
40 views
Prove this class of matrices has only one positive eigenvalue
I am working with real symmetric non-negative matrices with integer elements and zero diagonal. They are particularly nice, and I am fairly sure that they all have exactly one positive eigenvalue.
...
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votes
0answers
35 views
Determinant of matrix formed from blocks of a $2 \times 2$ block partitioned symplectic matrix.
I am working on a problem in quantum optics in that context, I came across the following determinant of a complex matrix of size $n \times n$ :
$$\mathbb{G}=\det\left[\mathcal{U}_{11}^{}+\mathcal{U}_{...
1
vote
1answer
37 views
Prove that these matrices are conditionally negative definite
I am working with real symmetric non-negative matrices with integer elements and zero diagonal. They are particularly nice, and I am sure that they are conditionally negative definite. That is, if $A$ ...
1
vote
1answer
75 views
Closed form solution for block matrix A in $AXA^T=C$
Looking for help for closed form solution for A, given C and X in the matrix equation
$AXA^T=C$
A (size $M x N$), X ($N x N$) and C ($M x M$) are (appropriately sized) block matrices
$A =
\...
1
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0answers
33 views
Trace of the square root inside commutation property
Hi I'am looking forward to prove this statement, which seems to be true (checked numerically) even if I cannot find it anywhere and didn't manage to prove it :
$\forall B\in M_{n,r}(\mathbb{R}),Tr(\...
2
votes
1answer
41 views
Graham's formula for determinant of distance matrix of a tree
The following is a proof of Graham's formula for the determinant of the distance matrix of a tree in the book Graphs and Matrices by R.B. Bapat.
I am not able to come up with the final matrix $D_2$ ...
1
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1answer
38 views
Does maximizing the Rayleigh quotient using the method of Lagrange multipliers require the matrix to be positive semidefinite?
I’m faced with the problem of maximizing a Rayleigh quotient:
$$\max_h \,\, \frac{h^t H h}{||h||^2}$$
Which is equivalent to solving
$$\max_h \,\, h^t H h $$
$$ s.t. ||h||^2=k >0, k \in \...
-4
votes
1answer
146 views
Are the corner hypercubera polytopes self-dual?
Motivation: The polyhedron whose vertices are seven of the vertices of a cube (four on the bottom and three on top) - called a cubera - is self-dual. Does an analogous construction produce a self-dual ...
4
votes
1answer
36 views
Proving $S= S^3$
Let $\mathrm{A,B,C,D}$ be (not necessarily square) real matrices such that $\mathrm{A^T = BCD , B^T= CDA, C^T = DAB, D^T = ABC}$ for the matrix $\mathrm{S= ABCD}$ prove that $\mathrm{S= S^3}$
Attempt:...
2
votes
3answers
74 views
Prove that the span of $\{M_1, M_2, M_3\}$ is the set of all symmetric $2\times2$ matrices.
From Linear Algebra by Friedberg, Insel, and Spence:
Given $M_1=\begin{pmatrix} 1&0\\0 &1\end{pmatrix}$, $M_2=\begin{pmatrix} 0&0\\0 &1\end{pmatrix}$ and $M_3=\begin{pmatrix}0&1\...
1
vote
1answer
52 views
Rank property of a matrix including symmetric and persymmetric Hankel matrix
I am investigating the property of the Hankel Matrix recently. Here, I have a question, what is the rank property of $[\mathbf{S}\ \mathbf{P}]$, where $\mathbf{S}$ is a symmetric hankel matrix and $\...
1
vote
1answer
37 views
equivalent statement for positive definitiveness
In this lecture (https://engineering.purdue.edu/~ragu/confpapers/Bal302-talk.pdf) page $14$, it is said that for a real matrix $A$ and a symmetric positive definite matrix $P$ the statement
$$ A^T P + ...
