Questions tagged [symmetric-matrices]

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

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Relation between symmetric outer product decomposition and symmetric multilinear decomposition

Suppose tensor $\mathcal{A}$ is a symmetric real tensor of order $k$. Then, symmetric outer product decomposition of $\mathcal{A}$ is $$ \mathcal{A} = \sum_{i=1}^p \lambda_i v_i^{\bigotimes k}, $$ ...
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How to estimate the spectralnorm by the infinity norm?

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric, positive-definite matrix. Furthermore let $A$ be diagonally dominant, i.e. $\max_{i=1,...,n} \sum_{j=1, j\neq i}^{n} \frac{|a_{ij}|}{|a_{ii}|} \leq ...
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Does symmetry of $AB$ implies symmetry of $A^\dagger B$?

Let $A$, $B$ and $AB$ symmetric. Is $A^\dagger B$ also symmetric i.e. $$A^\dagger B = B A^\dagger$$, where $A^\dagger$ is the pseudo-inverse of $A$ ?
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Does $XWX + WXX = 2WX$ lead do $X=I_n$?

Let $X,W\in\mathbb{R}^{n\times n}$ be square and symmetric real matrices. Does the fact that $$ XWX + WXX = 2WX $$ lead necessarily to $X=I_n$? Obviously $X=I_n$ satisfies the above equation, but does ...
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Conditions on 3 x 3 matrix elements for the matrix to be positive semi-definite by strictly using the definition of positive semi-definite matrices.

A matrix $M$ is considered positive semi-definite if $$ z^{T}Mz \geq 0 $$ Using this definition, I am trying to discover all conditions required on the elements of the 3x3 symmetric matrix $M$ for it ...
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When does a symmetric matrix admit a positive eigenvector (i.e. $v_i >0 \forall i$) for a positive eigen?

Let $S \in \mathbb{R}^{n \times n}$ be a symmetric matrix. I am trying to understand when such a matrix possesses a positive eigenvector for a positive eigenvalue, that is a $v \in \mathbb{R}^n, v_i &...
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Is a scalar product positive definite on a unique maximal subspace?

Let $V$ be an $n$-dimensional real vector space and \begin{equation} \eta\colon V\times V\to\mathbb R \end{equation} a nondegenerate symmetric bilinear form. Sylvester’s Law of Inertia allows us to ...
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Euclidean projection on convex set of positive semidefinite matrices

Define the Euclidean projection for a convex set $C$ as follows $$\pi_C(y) := \min_{x \in C} \| y - x \|_2^2$$ How would we find the projection map when $C$ is the cone of positive semidefinite ...
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Proof that $(A^t)^t=A$, $(A+B)^t=A^t+B^t$, $(AB)^t=B^tA^t$, and deduce that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric

In a linear algebra textbook, I was given the following problem: If $B$ is a $n \times n$ square matrix, show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric. I know that there are relatively ...
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Decomposition of order 3 tensor symmetric along two dimensions

I have a 3rd order tensor $\mathbf{A}$ consisting of symmetric covariance matrices (with dimensions of space by space) stacked in time. I would like to compute the leading spatial features that ...
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If $A$ is a symmetric matrix, then $\det(A) \leq \prod\limits_{i = 1}^d a_{ii}.$

Prove or provide a counterexample. If $A = (a_{ij})$ is a symmetric matrix, then $\det(A) \leq \prod\limits_{i = 1}^d a_{ii}.$ The result is obviously true for diagonal matrices and here is a proof ...
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Prove the norm-preserving matrix extension theorem $\min\Vert\binom{A~C}{B~W}\Vert_2=\max\{\Vert\binom{A}{B}\Vert_2,\Vert(A~~C)\Vert_2\}$ by symmetry?

I want to prove the following norm-preserving matrix extension theorem: Given matrices $A\in\mathbb C^{k\times k},B\in\mathbb C^{(n-k)\times k},C\in\mathbb C^{k\times(n-k)}$, then $$\min\bigg\Vert\...
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Is a map which sends a $3\times 3$ symmetric tensor to an element of $SO(3)$ which diagonalizes it necessarily discontinuous?

For a $3\times 3$ symmetric matrix $Q$, one can construct a map to $SO(3)$ which sends $Q$ to a matrix which diagonalizes it. If $Q$ has distinct eigenvalues, there are three choices for rotation ...
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Smallest eigenvalue of a nearest neighbor matrix in $2$ dimensions.

Consider a 2D square lattice with $n \times n$ lattice sites. A matrix $M_n$ of size $n^2 \times n^2$ is constructed by setting $M_{ij} = u$ (where $0 \leq u \leq 1$) if sites $i$ and $j$ are nearest ...
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orthogonal polynomials and determinant of jacobi matrix

In the book 'Orthogonal Polynomials of Several Variables' by Charles F. Dunkl and Yuan Xu page 11 is the picture below. I assume the matrix in the picture is related to Corollary 1.3.10 For the case ...
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The eigenvalues of a matrix composed of powers

During my research work related to the convergence of estimates, I needed to calculate the eigenvalues of the following symmetric matrix. $$\Sigma_{n\times n} := \begin{pmatrix} 1 & \frac{a}...
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Maximizing generalized Rayleigh quotient with constraints

Let $A$ and $B$ be $n \times n$ symmetric matrices with real entries and let $k \geq 2$ be an integer. I want to find the maximum of $$ \frac{\sum_{i=1}^k X_i^{\mathrm T}\,A\,X_i}{\sum_{i=1}^k X_i^{\...
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Prove that there is a positive real number $\lambda$, so that $A =\lambda B$, for two positive definite square matrices [closed]

A and B are positive definite square $n$ $\times$ $n$ matrices. The thereby defined dot products define the same orthogonality relation. Which means: $∀v, w ∈ R^n : v · Aw = 0 ⇔ v · Bw = 0.$ Show that ...
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Is the space of $n\times n$ real symmetric matrices with strictly positive determinant connected within the vector space of $n\times n$ real matrices?

I want to make clear that I am aware of the connectedness in the case of general real matrices. But here I ask about the subspace of symmetric ones. If it is not the case, which are the connected ...
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Some questions about the conditions for the CARE/DARE solutions

I'm trying to understand the CARE/DARE solution and have a few questions. For example, for the CARE the equation is $$A^tP + PA - PBR^{-1}B^tP+Q=0$$ where $P,Q,R,A,B$ are $n\times n$ matrices and $P,Q,...
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Bound on the trace of a product of a symmetric positive definite matrix and a nil-potent matrix.

Let $A$ a matrix with the following form: $A = vu^{T}$ where $u$ and $v$ are orthogonal vectors and $B$ be a symmetric positive definite matrix. Is there a way to upper bound $Trace(AB)$? I know that ...
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Determine all real $2 \times 2$ matrices that commute with their transpose

Determine all real $2 \times 2$ matrices $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ that commute with $A^T$: $$A \cdot A^T = A^T \cdot A$$ A matrix commutes with its ...
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Product of symmetric matrices equals Zero matrix, i.e., $X Z = 0_{n,n}$

Let $X \in \mathbb{R}^{n \times n}$ and $Z \in \mathbb{R}^{n \times n}$ be symmetric matrices. Assume that $Q=\left[q_{1}, \ldots, q_{r}\right]$ is an $n \times r$ matrix whose columns form an ...
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How I can convert a negative definite matrix to a positive definite?

If I have a negative definite matrix $A$, can I convert this matrix to a positive definite by taking columns $A$ of the negative eigenvalues?
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If A is symmetric and $A^2 = A$, show it is a projection matrix.

I assumed showing $A=QQ^T$ was enough to say it is a projection matrix, but I think it's not. I showed this by saying $A=QDQ^{-1}$ (D is diagonal) and knowing $Q^T = Q^{-1}$ for orthogonal matrices, ...
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Spectral properties of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)}~~~~~~~~~~~~~~~...
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Universal Properties of Orthogonal Matrices

I wanted to ask this question since I have seen conflicting viewpoints on it. Are orthogonal matrices necessarily symmetric? I do not believe so but some website said they were so I need to confirm. (...
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The square of a real skew-symmetrix matrix is symmetric matrix

Let $S \in \mathcal{M}_n(\mathbb{C})$ be a real symmetric matrix. Show that $S$ is the square of a real skew-symmetrix matrix then all its eigenvalues are negative and its non-positive eigenvalues ...
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Conditions for a block matrix with parameters to be positive definite

Let $A$ and $B$ be $n \times n$ positive definite matrices. Let $C$ be any $m \times n$ matrix. Can we always find $\alpha$ and $\beta$ such that the matrix $$\begin{bmatrix} A + \alpha C^\top C + \...
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Eigenvalues of symmetric part of product of matrices

Consider a real matrix $B$ defined as $$ B := X A + A^T X $$ where $X$ is a symmetric positive definite matrix and $A$ has eigenvalues with positive real parts. How can I prove that eigenvalues of $B$ ...
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Nonsingularity of sum of matrices

Consider the matrix $E$ where $E$ is defined as follows: $E=A+\gamma BC+DB-BC^{-1}DC$ where $\gamma$>0, $A,C$ are positive definite, $D$ is Hurwitz and $B$ is positive semidefinite. I have a ...
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What is the correct representation of the generalized gamma function?

The NIST Digital Library of Mathematical Functions defines the multivariate gamma function as $$ \Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr% }\left(-\mathbf{X}\...
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Non positive-definiteness of some binomial matrices

Experience shows that the following matrix $$ A_4 = \begin{pmatrix} 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 3 & 4 & 0 \\ 1 & 3 & 6 & 0 & 0 \\ 1 & 4 & 0 &...
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A Geometrical Derivation of Quantum Mechanics Spin Operators

I'm trying to see if there is a way to geometrically derive a general form for the quantum mechanics spin operators. I'm trying to deduce their commutation relations without using any knowledge of ...
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Frobenius norm inequality for product of matrices with kronecker product structure

Consider three integers $p,n,t$. Consider a matrix $M\in R^{pt\times pt}$ symmetric positive definite with eigenvalues at most one of size $(pt\times pt)$, a matrix $A\in R^{t\times t}$ and a matrix $...
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What can I get if I choose a vector from a given set that spans the space to maximize the determinant of rank-one perturbations of a matrix?

We have a real symmetric positive semidefinite matrix $M_0 \in \Bbb R^{d \times d}$ and a given set $K \subset \Bbb R^d$ that contains $|K|$ $d$-dimensional vectors, and it spans $\Bbb R^d$. At each ...
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How can I prove (or disprove) that there is no matrix $C$ such that $C^T = C^{-1}$ and $A = C^{-1} B C$?

$A$ and $B$ are square matrices of the same size. $A$ is symmetric, and $B$ is not symmetric. How can I prove (or disprove) that there is no matrix C such that $C^T = C^{-1}$ and $A = C^{-1} B C$? I ...
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Prove convexity with hessian matrix

I've got a function $$f(x)=(x_1-1)^2+\sum_{i=2}^n (x_i-x_{i-1})^2\quad \text{with $x\in \mathbb{R}^n$}$$ I want to show that this is (strictly) convex, so I thought the best approach might be to look ...
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If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric?

If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric? I know that given the symmetric matrices $A$ and $B$, then $...
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What functions preserve symmetry and positive-definiteness of covariance matrices?

Suppose I have covariance matrices $H_1,…,H_n$ (symmetrical, positive-definitive) and corresponding weights $w_1,..,w_n$. We want to find a function $f$, such that $H = f(H_1,…,H_n,w_1,…,w_n)$ is ...
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Analytical solution to a matrix equation

I am trying to solve the following matrix equation $(I-X^{-1}BX^{-1})(X-C)=0$ where $X,B,C$ are positive definite and $X$ is unknown. Two obvious solutions are $X=C$ and $X=B^{\frac{1}{2}}$. We also ...
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Calculating Skew Symmetric Matrix determinant?

I've been solving matrix related questions and I'm confused for this one: If A is a 5 * 5 matrix, where A^t= - A (A transpose equal to negative A) then what is the determinant for A? There are 4 ...
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Is it always possible to split a square matrix into a product of two matrices [closed]

Can we always split a square matrix into a product of two matrices and if so, is there a standard procedure to do so? Please provide any known theory regarding this, even if it exists only for a ...
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Boyd and Vandenberghe Example 3.10: Convex function example

I am trying to understand how the function $$f(X) = \lambda_ max(X)$$, with dom(f) = Sm is convex. The authors say that $f(X)$ can be expressed as $f(X)$ = sup{$\\{y}^\intercal Xy$| norm(y)=1} is ...
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Doubts on symmetric and a skew-symmetric matrix

Given two following statements: $1.$ "The diagonal elements of a skew-symmetric matrix are all zero." $2.$ "A real/complex square matrix can be uniquely expressed as the sum of a ...
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Invertiblity of sum of matrices

Assume two matrices A and B are both symmetric positive definite. I was wondering whether I+AB and I-AB are invertible, where I is the identity matrix. (A hint might be AB is positive definite iff AB ...
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What's the singular value of a symmetric matrix plus identy matrix? $A+\lambda I$

Suppose, we know the singular values of a symmetric matrix $A$ as $\{\sigma_1,\cdots,\sigma_n\}$. What is the singular values of the matrix $A+\lambda I$?
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how to prove $(A^2)^+=M^2$ if $M=A^+$ and A is a symmetric matrix?

Here $A^+$ is Moore–Penrose inverse. My question is: If $M=A^+$ and A is a symmetric matrix,then how to prove $(A^2)^+=M^2$?
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Prove that if $B$ is symmetric then its matrix is symmetric.

Let $V$ be a finite-dimensional vector space over the field $F,$ and let $B$ be a bilinear form on V. $(a)$ $B$ is said to be symmetric if $B(v_1, v_2) = B(v_2, v_1)$ for all $v_1, v_2 \in V.$ Prove ...
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Set of symmetric $n\times n$ matrices $S_n$ is convex

I'd like to know if the set of symmetric $n\times n$ matrices $S_n$ is convex. I looked at the definition of a convex set and it says that a set $X$ is convex if $$\forall x,y \in X \, \forall \lambda ...
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