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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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Quadratic form of a real symmetric matrix is bounded

If $\lambda_1>\lambda_2>...>\lambda_r$ are the different eiegenvues of a real symmetric matrix $A\in M_{n\times n}(\mathbb{R})$. $1.$ Show that the quadratic form associated to $A$ satisfies ...
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Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices

Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2 ,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume ...
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Triviality in the proof of the spectral theorem for real matrices which I can't get my head around [closed]

I asked the following to chatGPT: Can you prove that every symmetric real matrix has an orthonormal basis consisiting of eigenvectors? - To which it responded with: Certainly! To prove that every ...
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Reasoning for reduced SVD factorization

I am aware that for any $m \times n$ matrix $A$, we can write: Known 1: $A = U\Sigma V^T$ where $U$ is orthogonal and $m \times m$, $V$ is orthogonal and $n\times n$, and $\Sigma$ is diagonal and $m \...
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Rank of a matrix $A$

Let $$A=\begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{13} & a_{14} & a_{24} \\ a_{13} & a_{14} & a_{24} & a_{34} \\ a_{14} & a_{24} & a_{...
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$A^3=A$, ${\alpha }^{T}{A}^{T}{A\alpha } \leq {\alpha }^{T}\alpha$ implies $A$ is symmetric?

Let $n\times n$ real matrix $A$ satisfies ${A}^{3} = A$; and for any $\alpha\in\Bbb R^n$, we have ${\alpha }^{T}{A}^{T}{A\alpha } \leq {\alpha }^{T}\alpha$, prove that $A$ is symmetric. My attempt: $...
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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 ...
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Is there such things as an inversion polynomial?

In enumerative combinatorics there is a descent polynomial that counts the number of permutations in the symmetric group of size n with a descent set equal to some fixed set S. I was wondering if ...
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Is it true that $D A P D^T = A D P D^T$ if $P$ is symmetrical, positive definite and $D$ is diagonal?

I know in general, matrix multiplication is not commutative, but would it be true in this special case? $D A P D^T = A D P D^T$ where $A, D, P$ are all $n by n$ matrix. But $P$ is symmetrical and ...
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How to decompose the Hessian matrix of a 3rd degree polynomial into 2 or more vectors/matrices?

I am trying to figure out what can be said about the spectral radius of the Hessian of a 3rd degree random polynomial defined over a unit hypercube, drawing on known results in random matrix theory. ...
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Determine if inner product over a real vector space has a certain form

Verify if the following statement is true: Every inner product on $\mathbb{R}^n$ has the form $\langle v,u\rangle = v(Au),$ where $A$ is a symmetric matrix with positive entries on the diagonal. I ...
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Find values which make two matrices congruent

For which values of $a\in\mathbb{R}$ are the following matrices congruent? $$A=\begin{pmatrix} 1&4-a-a^2\\ 2& -1 \end{pmatrix}$$ $$B=\begin{pmatrix} -a-1 & 3\\ 3 & -5 \end{pmatrix}$$ ...
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Fastest way to divide by a symmetric positive matrix

Say $P_{yx}$ is a general $(n_y,n_x)$ matrix. Say $P_y$ is symmetric, positive definite, of size $(n_y,n_y)$. I want to compute $GT=P_y\backslash P_{yx}$ (matrix left division, or perhaps more ...
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For a real symmetric matrix, is the product of two factors of its rank decomposition (right times left) also symmetric?

Recently, I am learning generalized inverse of a matrix. Given a real symmetric matrix $A\in\mathbb{R}^{n\times n}$ with ${\rm rank}(A)=r$, suppose the rank decomposition of $A$ is given as follows: $$...
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A matrix inequality $X - \frac{1}{u^\top Xu} Xuu^\top X \succeq Y - \frac{1}{u^\top Yu} Yuu^\top Y$

Problem. Let $n\ge 2$. Let $X, Y$ be $n\times n$ real symmetric positive definite (PD) matrices with $X \succeq Y$ (i.e. $X - Y$ is positive semi-definite (PSD)). Let $u\ne 0$ be a $n\times 1$ real ...
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For a symmetric matrix $B$ and following four relevant matrices $P,Q,C,D$, what's the relation between $QP$ and $CDC^{\rm T}$?

Suppose $B\in\mathbb{R}^{n\times n}$ is a symmetric matrix with ${\rm rank}(B)=r$. Then $B$ is equivalent to $\tilde{B}$ in (1), where $I_{r}$ denotes the identity matrix of order $r$. That is, there ...
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If $B$ is PD in null space of $A\in \mathbb{R}^{m\times n}$ then exists $r \in \mathbb{R}$ such that $B+rA^TA$ is PD

If a symmetric matrix $B$ is PD in null space of $A\in \mathbb{R}^{m\times n}$, with rank$A=m<n$, then exists $r \in \mathbb{R}$ such that $B+rA^TA$ is PD. What I've tried is the following: If $x\...
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A question about permutation similarity of symmetric matrices

Let $A , B \in M_{4 \times 4}(\mathbb{R}_{\geq 0})$. Matrices $A, B$ are $\textbf{permutation similar}$ if there exists a permutation matrix such that $A= PBP^T$. Define $\mathrm{diag}(A)$ to be the ...
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$E$ is finite-dimensional Euclidean space over $\mathbb{R}, x \in E, x \neq 0$. Then $\{Ax: A = A^* \succeq 0, \|A\| \leq 1\}$ is a closed ball

Let $E$ be a finite-dimensional Euclidean space over $\mathbb{R}$ and let $x \neq 0$ be a vector in $E$. Show that the set $K=\{Ax: A = A^* \succeq 0, \|A\| \leq 1\}$ is the closed ball of radius $\...
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Equivalence for a matrix to be symmetric

Let $A\in\mathbb R^{n\times n}$ with the QR-decomposition $A=QR$ ($Q$ orthogonal, $R$ upper triangular matrix). I asked myself if the following statement holds: $$A\in\mathbb R^{n\times n}\text{ ...
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Doubts about geodesics equation derivation from Lagrange's equations

I want to derive the geodesic equation from a Lagrangian pov, so I consider a Lagrangian $L(q,\dot{q})$ given only by the kinetic energy wrote as the quadratic form of the kinetic matrix $A(q)$, i.e (...
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$A-B$ semi positive definite can tell us the information of the positive inertia index

Suppose $f,g$ are symmetric real quadratic forms on $\mathbb{R}^n$ such that $f(x)\geq g(x)$ for all $x\in\mathbb{R}^n$. Prove that the positive inertia index of $f\geq$the positive inertia index of $...
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Inverse and Determinant of Matrix $Axx^TA+cA$

Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
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Largest singular value of anti-triangular matrix

I have an anti-triangular matrix that is not Hermitian (it is complex and symmetric). I would like to find a method to bound its largest singular value. Might that exist, in general? Anti-triangular ...
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How to compute Sylvester form of a matrix representing a symmetric bilinear form?

Can somebody state a step-by-step algorithm to, given a symmetric n x n-matrix A, (congruently) diagonalize A such that the entries of the diagonal are 1, -1 and 0 corresponding to the signature of A ...
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Modified graph Laplacian, D + A

Consider an undirected, connected graph with positive edge weights $G$ with adjacency matrix $A$ and (diagonal) degree matrix $D$, and graph Laplacian given by $L = D - A$. $L$ is singular and is non-...
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Singular values as min max of absolute rayleigh quotient

Consider a real symmetric matrix M, satisfying $\mathbf{1}^\intercal M = \mathbf{1}^\intercal$ having eigenvalues $1=\lambda_1 \gt\|\lambda_2\| \geq \|\lambda_3\| .... \geq \| \lambda_n\|$, then can I ...
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Is $l_1$-norm isometrically embeddible into Hilbert space?

I am reading Schoenberg's article "METRIC SPACES AND POSITIVE DEFINITE FUNCTIONS". There he proves that $K({\mathbf x}, {\mathbf y}) = e^{-\gamma\|{\mathbf x}-{\mathbf y}\|_p^q}$ is a ...
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Lyapunov Special Symmetric Case [closed]

Consider the Lyapunov equation: $$AX+XA = B$$ and assume that $A$ is symmetric positive definite and $B$ is symmetric. I am not able to proof that $X=X^{T}$ holds. Would be grateful if somebody could ...
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Proving rank of $A-\lambda vv^t $ is $r-1$.

Let $A$ be an $m\times m$ symmetric real matrix of rank $r$ s.t. $r\ne m$.If $\lambda$ nonzero is an eigenvalue of $A$ with corresponding unit column vector $v$ s.t. $Av=\lambda v$.Then prove the ...
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Writing convention of Courant–Fischer theorem

Let $A \in \mathcal{M}_n(\mathbb{R})$ be a symmetric matrix and $\lambda_1 \leq \lambda_2\dots\leq\lambda_n$ be its real eigenvalues taken with multiplicities. Let $1\leq i_1 \leq i_2\dots\leq i_k\leq ...
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Why $X^\perp=\text{span}\space\{x_{k+1}\dots,x_n\}$ true?

Given the set $X=\{x_1,x_2,\dots\,x_k\}$ be a orthonormal set of eigenvectors of a symmetric matrix $A\in\mathcal M_n(\mathbb{R})$. Then I don't understand why $X^\perp=\text{span}\space\{x_{k+1}\dots,...
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Different versions of Courant–Fischer theorem

Let $A \in \mathcal{M}_n(\mathbb{R})$ be a symmetric matrix and $\lambda_1 \leq \lambda_2\dots\leq\lambda_n$ be its real eigenvalues taken with multiplicities. Let $1\leq i_1 \leq i_2\dots\leq i_k\leq ...
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Prove that if $\lambda$ is an eigenvalue of a symplectic matrix, then $\frac{1}{\lambda}$ is also an eigenvalue of such matrix

I´m trying to solve the following problem: A symplectic $n\times n$ matrix $A$ follows this conditions: $J$ is a $n\times n$ matrix $J^2=-I$ $A^TJA=J$ $n$ is an even number Prove that if $\lambda$ ...
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Example of semi algebraic subset over a real closed field

I am trying to find interesting example of semi algebraic subset over a real closed field other than those we have by considering $\mathbb{R}^n$. I tried to construct an example involving matrices. ...
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A case problem about rank-1-perturbation of diagonal matrices

I have the following prediction for rank-1 perturbations of diagonal matrices, but I don't know how to prove (or disprove it). Given $v:= [v_1,...,v_K] \in (0,1]^K$, we define $a:= \sum_{k=1}^K v_k = \...
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Inequality involving a symmetric matrix and minors of an orthogonal matrix

Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...
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Degenerate spectrum for the sum of projections onto unit vectors that sum to zero implies a symmetry of the vectors?

I am interested in a sum of projections $A = \sum_{i=1}^N \mathbf{a}_i\,\mathbf{a}_i^T$, where $\mathbf{a}_i$ are real column vectors with unit norm and $\sum_i \mathbf{a}_i = \mathbf{0}$. So $A$ is a ...
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Limit of Eigenvalues of Matrix Expression

I would like to study the limit $$ \lim_{x\rightarrow \infty} \lambda_{\min} [x(I_n + xA)^{-1}] , $$ where $x \in \mathbb R_{\geq 0}$ is a scalar, $I_n$ is the $n\times n$ identity matrix and $A \in \...
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Solving Xa=b for an unknown matrix X

I'm interested in studying the solutions of $Xa=b$ for an unknown square matrix $X$, and given (known) column vectors $a$ and $b$ in $\mathbb{R}^n$. For any numerical $a, b$, one can directly attempt ...
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Given a symmetric diagonally dominant matrix, does $-|a_{ii}|\le \sum_{j\neq i} a_{ij}$ hold?

I am studying linear algebra and I am playing with some basic notion (symmetric matrices, pivots, diagonal matrices etc). Let $A\in\mathbb R^{n\times n}$ be a symmetric matrix and consider the ...
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There is a permutation matrix $P$ such that $PAP^{T}$ is in this form for symmetric $A$

Suppose that $A$ is a real matrix and is symmetric and nonzero, then I want to prove that there is a permutation matrix $P$ such that $PAP^{T}=\left[\begin{array}{ll} B & E^{\top} \\ E & C \...
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Inverting a specific symmetric matrix preserves its zero entries

Suppose $S$ is an invertible symmetric matrix with the following property: For the entry in the $i$th row and $j$th column, if $|i-j|$ is an odd number then $S_{ij} = 0$; if $|i-j|$ is an even number ...
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Proof verification: ''standard'' lower estimate for positive definite quadratic form [closed]

Let $Q(x_1,\ldots,x_n)$ be a positive definite quadratic form. I would like to show that there exists $C>0$ such that $Q(x_1,\ldots,x_n)\geq C(x_1^2+x_2^2+\ldots x_n^2)$. Proof: Let $\vec{x}=\begin{...
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How to show the quadratic form of the following matrix converges to zero?

Suppose we have a $l\times l$ real matrix $M=(X'X)^{-1}X'\Sigma X (X'X)^{-1}$, where $X$ is a $n\times l$ real matrix and $\Sigma$ is a $l\times l$ real symmetric and positive definite matrix. $X'X$ ...
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How to efficiently compute the determinant of a matrix with unknown diagonal entries?

I would like to ask Python to compute the determinant of a large symmetric matrix where all off diagonal entries are known. The diagonal entries could vary. Since I need to compute the determinant ...
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1 answer
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Solve $\| X A - B \|$ subject to $X C = C X$

Given $A, B \in \mathbb{R}^{n \times k}$ and S.P.D. $C \in \mathbb{R}^{n \times n}$, I would like to find an analytical solution for the matrix $X \in \mathbb{R}^{n \times n}$ that minimizes \begin{...
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Would $v^TAv>0$ forall $v$, $v$ if A's eigenvalues are all > 0? [duplicate]

I'm learning that if $n \times n$ positive definite matrix A A's then $\forall v$ $v^TAv>0 \ $, but this requires A is symmetric(then A have orthonormal eigenvectors) so I'm asking: 1.if A is not ...
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Symmetric linear least-squares solution with known diagonal elements

Given matrices $\pmb{A}\in\mathbb{R}^{p\times n}$ and $\pmb{B}\in\mathbb{R}^{p\times n}$ with $p>n$, I need to solve the following linear system in symmetric matrix $\pmb{X}\in\mathbb{R}^{p\times p}...
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Counting number of free parameters from a matrix constraint equation

How many free parameters does a $28 \times 28$ symmetric matrix $M$ have if it is subjected to a constraint $$M^T L M=L \tag{1}$$ where $$L=\begin{bmatrix} 0&I_6&0\\I_6&0&0\\0&0&...
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