Questions tagged [symmetric-matrices]
A symmetric matrix is a square matrix that is equal to its transpose.
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Help in proving the dimension of the symmetric linear maps
I'm trying to prove that the set $$S=\{A \in \mathscr{L}(\mathbb{R}^N,\mathbb{R}^N) | A^*=A\}$$
ie the symmetric linear maps in $\mathbb{R}^N$ is a vector subspace of $\mathscr{L}(\mathbb{R}^N,\mathbb{...
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condition to be a symmetric matrix
Let us consider two matrices $X$ and $Y$ where both $X$ and $Y$ are $n \times k$ and $X \neq Y$.
It is clear that $X'X$ is a symmetric matrix. What I want to know is whether $X'Y$ is always non ...
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Eigenvalues of $M^{\top} D M$ with $M$ skew-symmetric and $D$ positive diagonal.
I was wondering whether there was a way to bound the smallest eigenvalue of $M^{\top} D M$ when $M$ is skew-symmetric and $D$ is a positive diagonal matrix.
In particular, I was hoping that $M^{\top} ...
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Zero set of system of two real quadratic forms
Background: Consider the equation $x^T A_1 x = 0$ where $x \in \mathbb{R}^\mu$ and $A_1 \in \mathbb{R}^{\mu \times \mu}$ is a symmetric matrix. Suppose we also demand the normalization $x^T x = 1$. ...
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Effect of Scaling a Row and Column on Multiplicities of Nonzero Eigenvalues of a Symmetric Matrix
Question:
Let $A$ be a real symmetric $n\times n$ matrix with eigenvalues $\lambda_1, \lambda_2,\cdots, \lambda_n $ and corresponding algebraic multiplicities $a_1, a_2, \cdots, a_n$.
Consider a ...
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Determinant of a particular form of block matrix with commutating property
What is the determinant of the block matrix
$$\begin{pmatrix}A&J&\ldots&\ldots&J\\J&A&\ldots\ldots&J\\\vdots&\vdots&\ddots(2^m-times)&&\vdots\\\vdots&\...
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Determinant of sparse Hilbert matrix
It is known that the determinant of the Hilbert matrix of dimension $N$ with elements
$$
H^N_{tr}=\frac{1}{t+r-1}, \quad t,r=1,\dots,N
$$
namely of the form
\begin{pmatrix}
1 & \frac{1}{2} & \...
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Prove that if $A$ is symmetric and nonsingular, then $A^{-1}$ is symmetric
Here is my attempt:
If $A$ is nonsingular and $A=A^T$, then $A^{-1}A=A^{-1} A^T=I_n.$ Since $A^T$ is nonsingular, we multiply $(A^T)^{-1}$ on both sides and we get $(A^T)^{-1}I_n=A^{-1}I_n \implies (A^...
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Intuitive difference between optimal transport distance and Fisher information distance
Let me start by saying I'm not a mathematician but a biologist with an interest in mathematics.
I have a set of covariance matrices and I am interested in studying their geometry in the Symmetric ...
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Multiplication with a diagonal matrix that can have infinity values
I'm thinking how to program the evaluation of the expression $A (B+A)^{-1}$, where $A$ is a diagonal matrix with elements equal or greater than zero, and B is a square symmetric definite positive ...
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when A matrix has one solution IF M>N [closed]
if a matrix A is m by n , and m>n , in this case we have one solution? i don't understand how more rows of a matrix can still have one solution
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Does the product of a PSD matrix and a positive eigenvalue matrix have positive eigenvalues?
It's known that if $A$ is a positive definite (symmetric) matrix and $B$ is positive semi-definite (not necessarily symmetric), then $AB$ has non-negative eigenvalues. It follows from knowing that $A$ ...
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Solving a certain matrix differential equation
I am trying to solve the following differential equation. All the matrices here are on $\mathbb{R}$.
Let $\sigma$ be a time $t$ dependent $2n \times 2n$ Hermitian matrix. I want to solve the following ...
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suppose $A$ is an $m\times n$ real matrix and $A^\top A$ is $3\times3$ nonsingular matrix, then $n =$? [closed]
the answer of this question is $n<3$ , why? I mean how we decided that is $n<3$ when $A$ multiply $A^\top$ will be singular?
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A,B matrices and column space [closed]
if A,B have the same column space , does that means they have the same number of rows and the same rank?
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Invariant eigenvalues of real symmetric matrices under sign change of off-diagonal elements?
I am looking for the minimum additional restrictions on a real symmetric $(n\times n)$ matrix $A$, such that its eigenvalues are invariant under a sign change of all off-diagonal elements.
I don't ...
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Distance between two positive semidefinite matrices
Given two positive semidefinite matrices X, and Y. My question is, can we use von Neumann or the log-Determinant divergences to measure the distance between the two matrices. Most Manifold measure ...
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determinant of symmetric block matrix with positive definite diagonal blocks
I have a matrix of the form
$B = \left[\begin{array}{cccc}
A_1 & C^T\\
C & A_2\\
\end{array} \right]$
Where $A_1,A_2$ and $C$ are all square, and $A_1,A_2$ are symmetric positive definite. ...
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Proving that a symmetric real matrix with a specific structure is positive definite.
Let $H$ be an $N\times N$ symmetric matrix with the following structure:
\begin{equation}
[H]_{n,m}\triangleq\begin{cases}
{\left|x_{n}\right|}, &\text{if}\ n=m,\\
-\mathsf{Re}{\left(...
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Invertibility of the symmteric matrix observed under least squaress solution
I am having trouble in proving the following statement
Let $X$ be a $K \times d$ matrix with rank $d$ and let $W$ be a diagonal matrix with $d$ non-zero positive elements on the diagonal, that is rank ...
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Proving that the semi-implicit Euler method is symplectic
I'm having trouble understanding the following proof that the semi-implicit Euler method
$$\begin{cases} p^{k+1} = p^k-\Delta t \frac{\partial H}{\partial q}(p^{k+1},q^k) \\ q^{k+1} = q^k + \Delta t \...
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Understanding an Inequality In a Paper
The paper is the following:
https://arxiv.org/pdf/1608.06412.pdf
Right above (5) on page 5, the first inequality of the line where they use lemma 8, I'm not quite sure I see how they are using lemma 8....
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What's the best way to make a symmetric matrix positive definite?
Assume that you have a matrix $X \in \mathbb R^{m \times m}$ and it's symmetric, but it's not positive definite.
What's the best way to turn the matrix $X$ into a positive definite matrix?
I have a ...
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On the $\log \det$ of identity matrix plus a symmetric positive definite matrix
I am trying to learn some matrix differentiation, and came across example of calculating the derivative of $$f(X)=\log\det(X)$$ where the $X$ is a symmetric positive definite matrix.
I came to the ...
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Is there a case in which PSD matrices $A$ and $B$ satisfy the condition $A \approx B$ but $A^2 \not \approx B^2$?
For two PSD matrices $A, B$ and a positive number $\alpha$, let's define $A \approx_{\alpha} B$ as $A \preceq \alpha B$ and $B \preceq \alpha A$, where $A \preceq B$ means $B - A$ is PSD. And if there ...
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Existence of a real symmetric matrix similar to a Jordan block corresponding to 0
Let $n$ be a positive integer. Let $J_n(0)$ denote the $n$-by-$n$
Jordan block corresponding to 0.
For each $n$, does there exist a real symmetric matrix $B$ such that $B$ is
similar to $J_n(0)$?
...
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If $A$ is positive real symmetric matrix and $B$ is real symmetric, does there exist some $V$ such that $V^{T}AV$=$I$ and $V^TBV$ is diagonal matrix?
Let's say we have two matrices $A$ (a positive real symmetric matrix) and $B$ (a real symmetric matrix). And let us suppose that in general $A$ and $B$ don't commute with each other. Then,
Q. Is it ...
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$A,B$ positive semidefinite matrices with $A \geq B$ implies $A^2 \geq B^2$? [duplicate]
Is it true that if I am considering positive semidefinite matrices $A, B$ with $A \geq B$ then $A^2 \geq B^2$?
Could you help me prove this or think of a counterexample (eventually assuming the ...
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Inequalities about trace of Hadamard product of matrices
Denote $A\circ B$ as the Hadamard product of two matrices, that is, $$A\circ B=(a_{ij}b_{ij}).$$ Let $A$ be a $n\times n$ symmetric positive definite matrix. First, I know that $$tr(A\circ A)\leq tr(A^...
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When is a symmetric matrix with diagonal 1 and off-diagonal <1 positive definite?
I'm interested in running simulations with multivariate normal distributions in which the covariance matrix A has diagonal entries $a_{ii}=1$ and off-diagonal entries $|a_{ij}|<1,i\neq j$. There ...
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Connection between multiplication table $n * k$ and partial sums of the partial sums of the Dirichlet inverse of the Euler totient function.
I am watching this video: L-functions and the Langlands program (RH Saga S1E2)
This reminds me of a recurrence:
Let $c=1$ and a recurrence be:
...
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On the transitivity of the Löwner order
If the Löwner order is a partial order, then it is transitive. If so, how can one prove it?
Proposition. Let ${\Bbb S}_n ({\Bbb R})$ denote the set of $n \times n$ symmetric matrices over $\Bbb R$. ...
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Eigenvalues of a symmetric tridiagonal matrix
I'm looking for the eigenvalues of the following symmetric tridiagonal matrix
\begin{pmatrix}
a & z & 0 & 0 & 0 \\
z & b & z & 0 & 0 \\
0 & z & 0 & z & ...
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Matrix relations involving trace and eigenvalues
Given the real symmetric positive definite matrix $A \succ 0$, consider the following inequality in $\alpha \in \mathbb{R}$.
$$ A \succ \alpha I $$
where $I$ is an identity matrix of appropriate ...
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Inequality involving quadratic forms for positive definite symmetric matrices
Suppose that for a given fixed $v$, and two positive definite symmetric matrices $A$ and $B$ (with real-valued entries) it holds that:
$$
v^T A v > v^T B v.
$$
Does it then hold that
$$
v^T A^{-1} ...
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If $A$ is symetric and positive definite, when will $A + B$ be invertible?
$A$ is a symmetric and positive definite matrix with size of $n \times n$, and $B$ is a matrix of the same size. Under which circumstance will $A+B$ be invertible?
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Trace of square of a rank-$1$ Hermitian matrix ${\bf A} = {\bf a}{\bf a}^H$
Given matrix ${\bf A} = {\bf a}{\bf a}^H$, where ${\bf a}$ is a complex column vector. Are the following correct?
${\rm Tr}({\bf A}) = {\rm Tr} \left( {\bf a}{\bf a}^H \right) = {\rm Tr}({\bf a}^H{\...
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Eigenspace and structured matrices
I have to design a matrix A that solves a linear system:
$y = A x$
The requirement is that A is an RBF kernel, i.e., it has the following structure:
$ a_{ij} = \alpha \exp\left (\frac{-d_{ij}^2}{2\...
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votes
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Iterative algorithm for computing $\Sigma^{1/2} x$
Say I have a PSD matrix $\Sigma$ and a vector $x$, is there an iterative algorithm (faster than computing $\Sigma^{1/2}$ using Cholesky decomposition) for computing $\Sigma^{1/2} x$?
(In this ...
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Projecting a symetric matrix on a subspace of known kernel
Given a symmetric matrix ${\bf S} \in S_n(\mathbb{R})$ and linearly independent vectors ${\bf u}, {\bf v}, {\bf w} \in \mathbb{R}^n$, how can one numerically compute the projection of $\bf S$ onto the ...
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Why does Wolfram say this symmetric matrix has complex eigenvalues?
According to Wolfram, the following matrix has complex eigenvalues.
Symmetric matrices have real eigenvalues, so I’m not sure what I’m failing to understand. The matrix is the shape operator of the ...
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Prove that condition number of a matrix increases in its dimension
This is my first time asking a question here. Thus, I apologise in advance if it is not articulated correctly or something else turns out to be wrong with it. Before asking the question itself, ...
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2
answers
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How to calculate the differentiation of this expressions which contains matrix variable?
it is assumed that $\mathbf{X}\in\mathbb{R}^{n\times n}$ is symmetric, and
$\mathbf{Y}=\mathbf{M}^\textsf{T}{\mathbf{X}}^{-1}\mathbf{M}$, where $\mathbf{M}\in\mathbb{R}^{n\times m}$ has no special ...
2
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Eigenvalues of a squared symmetric matrix
In Page 185 here it says
... $M^2 y=\sigma^2y$. Since $M$ is symmetric, it follows that $y$ is an eigenvector of $M$ with eigenvalue $\pm \sigma$.
It seems to contradict the example here.
What am I ...
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3
answers
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Span of positive (semi)definite matrices
In this answer, Ben Grossmann said that $\operatorname{span}\{P:P \succ 0\} = \{A:A = A^T\} =: \mathcal S$. I am not sure, however, why this is true. Also, I guess $\operatorname{span}\{ P:P\succeq 0 \...
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definiteness of asymmetric matrices
The english wikipedia page requires a symmetric matrix to determine definiteness. ( A is an n × n symmetric matrix ...)
The german Wikipedia page on the definiteness of matrices states that a $n\times ...
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Definition of the (anti-)symmetric part of a tensor
Consider the following tensor:
$$
A_{ij} = B_{k,i}C_{k,j}
$$
which can be decomposed into its symmetric and antisymmetric parts: $A_{ij} = A_{ij}^s + A_{ij}^a$. I'm all confused as to how to express ...
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What are the minimal conditions for this matrix to be invertible?
Let $A>0$ be a $n \times n$ symmetric positive definite (SPD) and $B$ be an $n \times m$ matrix, then form the matrix $$ C = \left[\begin{array}{cc} A & B \\ B^T & 0\end{array}\right],$$ ...
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Grothendieck's inequality guarantee of relaxation for Semidefinite problem
I am struggling to understand a proof of theorem 3.5.6 in Roman Vershynin's High-Dimensional Probability
Theorem:
$$\text{INT}(A) = \max_{x_i = \pm 1 \text{ for } i = 1,\ldots,n} \sum_{i,j=1}^{n} A_{...
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How do I prove that for any $2\times2$ matrix $A$, $A^2$ can be written in the linear form $aA + bI$ where $a$ and $b$ are scalars? [closed]
How do I prove that for any $2\times2$ matrix $A$, $A^2$ can be written in the linear form $aA + bI$ where $a$ and $b$ are scalars?
I've tried letting the elements of A be $w$, $x$, $y$, and $z$ then ...
