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Questions tagged [symmetric-matrices]

A symmetric matrix is a square matrix that is equal to its transpose.

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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 ...
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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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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 \...
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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} & \...
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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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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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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 ...
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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: $$tr(BS)\ge tr(B)\det(S)$$
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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}&{}&{}\\ {}&...
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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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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 = ...
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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+...
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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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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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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, $A = \operatorname{inverse}(A) = A^T$, ...
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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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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 <...
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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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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 $...
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$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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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 ...
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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. ...
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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....
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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?
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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 ...
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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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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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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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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 ...
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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 $n$....
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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}-\...
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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 ...
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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 $AB=BA$.
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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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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 ...
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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 ...
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2answers
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“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 ...
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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 ...
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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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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}_{...
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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 $A$ ...
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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 = \...
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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(\...
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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 $D_2$ ...
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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 \...
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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 ...
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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:...
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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\...
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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 $\...
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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 + ...