Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [symmetric-matrices]

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

0
votes
1answer
20 views

Given an SPD matrix, any diagonal submatrix of full rank must be SPD.

I need help with the following proof: Given a symmetric positive-definite matrix, show that any diagonal submatrix of full rank must also be symmetric positive-definite. Thanks
1
vote
0answers
30 views

In what sense is the column space of a symmetric matrix equivalent to its row space?

I have an elementary question about linear dependence and column/row spaces. Suppose we have a matrix $A=\begin{bmatrix}1&2\\2&4\end{bmatrix}$, and we construct the matrix $B=\begin{bmatrix}1&...
2
votes
1answer
42 views

Diagonalization of symmetric matrices

Mas-Collel, Whinston and Green's Microeconomic Theory (3rd edition) asserts that any symmetric matrix can be diagonalized as following: What would be the proof of this?
0
votes
0answers
12 views

Sparse banded matrix diagonalization

I have a group symmetric, banded matrices of different orders that I want to diagonalize. The matrices are quite large, which is causing difficulty in obtaining a fast solution using the LAPACK ...
0
votes
1answer
32 views

Orbit of a symmetric matrix under orthogonal conjugation

Let $A\in M_n(\Bbb{R} )$ be a symmetric matrix. I want to find a general formula for the diagonals of the matrices of the form $g^{-1}Ag$, where $g\in O_n(\Bbb{R})$. Here is what I did : Since $A$ is ...
1
vote
1answer
28 views

The spectral decomposition of skew symmetric matrix

I have been studying the spectral decomposition of the matrices and figured out that it works for symmetric matrices but it wont work for the skew symmetric ones well, the sign of the final matrix is ...
2
votes
3answers
46 views

Symmetric matrices containing $\{1, 2, 3, …\}$ in each line

I am a bit curious about an exercise. I was supposed to prove that there is no matrix that has $1,2,3,4$ in each line and is symmetric. I did, by examination of all such matrices. Now, there is a ...
0
votes
0answers
26 views

Inequality involving determinant and matrices?

Here is the statement : Let $A\in \mathcal{S}_n^{++}(\mathbb{R})$ and $B \in \mathcal{S}_n^{+}(\mathbb{R})$ then we have the following inequality : $(\det(A+B))^{\frac{1}{n}}\ge (\det(A))^{\...
-2
votes
1answer
54 views

An $n \times n$ matrix $A$ is called skew-symmetric if $A^T = −A$. What values of a, b, c, and d now make the following matrix skew-symmetric? [closed]

Let $$ A=\left( \begin{matrix} d & 8a-c & 8a+2b \\ a & 0 & 8-5d \\ a+5b & c & 0 \\ \end{matrix} \right) $$ Let $$ A^T=\left( \begin{matrix} d & ...
2
votes
1answer
68 views

Can an orthogonal non-symmetric 3x3 matrix have 3 real eigenvalues?

I was wondering if a non-symmetric orthogonal matrix can have his 3 eigenvalues in the real numbers. All the 3 real eigenvalues orthogonal matrix i've found are symmetric. Can someone give me a ...
2
votes
2answers
27 views

Quadratic form vanishing at certain points

Let $A\in\mathbb{R}^{d\times d}$ be a symmetric matrix, and $X_1,\dots, X_n\in \mathbb{R}^d$ be vectors with $n>d$ (if more convenient, one can assume ${\rm span}(X_1,\dots,X_n)=\mathbb{R}^d$. ...
0
votes
0answers
19 views

Spectral radius of a hollow symmetric block matrix

Let $B$ be a $2 \times 2$ matrix. Suppose we have a $4 \times 4$ real symmetric matrix of the following form $$A = \begin{bmatrix} O_2 & B \\ B^T & O_2\\ \end{bmatrix}$$...
1
vote
1answer
36 views

product between symmetric matrix and transpose matrix

Suppose that B is a symmetric matrix and A generic matrix with $A^T$ its transpose, is it true that: $B*A = B*A^T $ ? i think that $B*A = (B^T)*(A^T) = B*(A^T) $ so is true And if A,B were 2 ...
3
votes
1answer
49 views

How to prove concavity of $\textrm{Tr}(\exp(H+\log X))$?

I am studying random matrices and in the notes that I am reading, the teacher uses Lieb's inequality without proving it, that is $$X \to \textrm{Tr}(\exp(H+\log X))$$ is concave on the set of ...
3
votes
0answers
46 views

What's the geometric interpretation of the square root of a matrix?

Question: If I have a matrix $A$, I know that its square root is a matrix that has the same eigenvectors as $A$ but its eigenvalues are the square roots of the eigenvalues of $A$. What does this ...
0
votes
1answer
41 views

Derivative of squared form

Let $\boldsymbol{Y}$ be a matrix with dimension $2 \times n$, $\boldsymbol{X}_c$ and $\boldsymbol{X}_r$ are matrices $n \times p$, $\Sigma$ is a nonsingular and symmetric matrix $2 \times 2$, $\beta_c ...
2
votes
3answers
32 views

Solutions of $X+X^t=tr(X).A$ in $M_n(K)$, where $A\in M_n(K)$ is given

I recently stumbled upon the following equation in $M_n(K)$, with $K=\mathbb{R}$ or $\mathbb{C}$, where $X$ is an unknown matrix in $M_n(K)$: $$X+X^t=Tr(X).A$$ Of course, if $X$ is antisymmetric, the ...
0
votes
1answer
31 views

How to create a Toeplitz matrix from a vector

I have to create a Toeplitz matrix of a suitable form from a given vector The vector is $\;\{x[0],x[1],x[2] ... x[L-1]\}$ The matrix is of the form \begin{bmatrix}x[0]&x[1]&x[2]&...&...
2
votes
2answers
69 views

Linear Algebra: Find four unit vectors in $\mathbb{R}^3$ with the same angle between each.

I need to determine the set of all angles $\theta$ such that there exists some four distinct unit vectors $\vec{v}_1, ..., \vec{v}_4 \in \mathbb{R}^3$ and the angle between any $\vec{v_i}$ and $\vec{...
1
vote
1answer
64 views

Eigenvalues of a Covariance Matrix with Noise

Imagine to have a covariance matrix $2\times 2$ called $\Sigma^*$. \begin{bmatrix}1+\sigma^2&\rho_{12}\\\rho_{21}&1+\sigma^2\end{bmatrix} I know that $\rho_{12} = \rho_{21}$ because it's ...
0
votes
1answer
30 views

How do I proof that the cluster covariance matrix is symmetric?

I was reading Fuzzy clustering with volume prototypes and adaptive cluster merging by Kaymak, U and Setnes, M. Here it is written the following equation \begin{equation} P_i=\frac{\sum_{k=1}^{N} u_{ik}...
0
votes
1answer
20 views

prove if a and b are n square skew symmetric matrices then AB is symmetric if and only if A and B commute. [closed]

please help me to prove this prove if a and b are n square skew-symmetric matrices then AB is symmetric if and only if A and B commute
0
votes
1answer
38 views

Find an orthogonal matrix that diagonalises the matrix A

a) Let v1 = (1, 1, 0) and v2 = (1, 0, 1). If the eigenvalue of v1 is 2, what is the eigenvalue of v2? I found the answer to this by proving that these eigenvectors are not orthogonal by computing ...
0
votes
1answer
18 views

What is the normalized graph matrix if the row-sum of proximity matrix is zero?

Let $S \in \mathbb{R}_{\ge 0}^{n \times n}$ be the proximity (or similarity) matrix of a graph, e.g. $$ S = \left[ \begin{matrix} 0 & 0.9 & 0.3 \\ 0.9 & 0 & 0.4 \\ 0.3 & 0.4 & ...
0
votes
0answers
15 views

To show an operator is symmetric

Suppose I want to show that the two operators $\mathcal{L}$ and $\mathcal{A}$ where they are respectively: \begin{align} &\mathcal{L} = \begin{pmatrix} -J & 0 \\ 0 & 0 \end{pmatrix} \...
0
votes
2answers
14 views

Relationship of spectral radius to matrix norm

I just read that for a real symmetric matrix, the matrix $(A)$ norm equals the spectral radius $(p)$ to the $n^{th}$ power : $||A||=p^n$. I don't think this is true, is it? If so, where does it come ...
0
votes
0answers
29 views

Derivation of eigenvalues of a symmetric matrix

Given a diagonal matrix $D$, we symmetrize matrix $W$ by the following transformation: $S \equiv D^{1/2} W D^{-1/2} $ Now, $S$ is a symmetric matrix which is diagonalizable as $S = X \Lambda X^T$ ...
1
vote
1answer
59 views

A metric on non negative symmetric matrix

Let $w=\{w(i,j)\}_{1\le i,j\le m}$ be an $m\times m$ symmetric matrix with non-negative real entries such that $w(i,j)=0$ iff $i=j$. Show that $$d(i,j)=\min\left\{\sum_{j=0}^{k-1}w(i_j,i_{j+1}):k\ge1,...
1
vote
2answers
42 views

Relation between the eigenvalues of symmetric $A$, $B$ and $A+B$ when $AB=(BA)^{T}$

Let $A$ and $B$ be two real symmetric matrices. If $AB=(BA)^{T}$ then, is there any relation between the eigenvalues of $A+B$ and eigenvalues of $A$, $ B$? If $A$ and $B$ be two real symmetric ...
0
votes
1answer
22 views

Positive trace (all diagonal entries are positive) implies semipositive definite?

I am working on a matrix with all the diagonal entries are strictly positive while every other entry is strictly negative. This matrix is symmetric as well. I want to show that this matrix is ...
1
vote
0answers
29 views

Find a unitary basis of the $\mathbb{R}$-vector space of $n \times n$ (complex) Hermitian matrices

The question is on the title ($n \in \mathbb{N}^*$). To be clear, unitary basis here means basis consisting of (complex) unitary matrices. I wonder this question because recently I've read about ...
1
vote
0answers
48 views

Can this matrix be negative definite?

Let $d = 12$ and $m = 6$, and denote by $0_n$ and $I_n$ the zero matrix and the identity matrix of size $n \times n$. Let $D_+ \in \mathbb{R}^{m \times m}$ be a diagonal matrix with positive ...
3
votes
1answer
79 views

Do matrices $\mathbf{A A}^H$ and $\mathbf{A}^H \mathbf{A}$ have the same eigenvalues?

Let $\mathbf{A}$ be any complex matrix. Do matrices $\mathbf{A A}^H$ and $\mathbf{A}^H \mathbf{A}$ have the same eigenvalues? Note: The matrix $\mathbf{A}^H$ is the conjugate transpose of the matrix $...
1
vote
2answers
50 views

Inversion of an almost identity matrix

Say we have a square matrix like so 1 c c c ... c c 1 c c ... c ... c c c c ... 1 What would be the inverse of this matrix? Calling an inv function is expensive, ...
0
votes
0answers
26 views

Optimization with a symmetric matrix constraint

I have a problem where I need to find the optimal $X\in S_{++}^n$ (i.e. $X$ is positive definite) for a strictly convex function $f(X)$. For what I understand, I need to assign a positive ...
0
votes
1answer
43 views

How to prove eigenvalues of specific block matrix are as proposed

In some of my work (statistics), I need the eigenvalues of a very large matrix. As such I would like to reduce it to a simpler problem and it seems entirely possible to me as the matrix has a very ...
0
votes
0answers
25 views

Let $A\in\mathbb{R}^{m\times n}$, and $Ax=b$ has a solution for each vector $b$, prove that

1) $m \le n$ 2) there is an inverse matrix $P$ so that $PA$ is a simplified matrix ( the one with all zeros except $1$'s diagonally) 3) there's a matrix $Q$ so that $AQ = I_{m}$ sorry for bad ...
0
votes
0answers
20 views

Field extension similarity

Let $F \subseteq K$ Be a field extension.I am Trying to prove that if $A,B \in M_n(F)$ are similar as matrices over the field $K$ (there exists an invertible matrix $P \in M_n(K)$ such that $PA=BP$) ...
3
votes
3answers
60 views

Eigenvalues of two symmetric $4\times 4$ matrices: why is one negative of the other?

Consider the following symmetric matrix: $$ M_0 = \begin{pmatrix} 0 & 1 & 2 & 0 \\ 1 & 0 & 4 & 3 \\ 2 & 4 & 0 & 1 \\ 0 & 3 & 1 & 0 \end{pmatrix} $$ ...
2
votes
0answers
36 views

Non-singularity of a certain block matrix

We are given a matrix of the form $$\left[\begin{array}{cccccccccccccc} 0&0&0&0&0&0&0&0&0&q_{1}&0&1&0&0\\ 0&2p_{2}q_{2}&0&0&...
1
vote
4answers
86 views

Find matrix $A$ given the matrix $X$ and that $X = AA^T$

I have a matrix $X = \begin{bmatrix}3 & 1\\1 & 1\end{bmatrix}$ and $X=AA^T$. How can I find $A$?
3
votes
2answers
40 views

Integral involving matrix exponential

Is there any way to simplify the integral $$ I = \int_{t_1}^{t_2}e^{\Lambda t} A e^{\Lambda t}\,dt $$ knowing that A is symmetric and Λ is a diagonal matrix?
0
votes
0answers
23 views

Trace of product of squared matrix and positive definite matrix is nonnegative (short exercise)

Let $\Sigma$ be a real $d \times d$-matrix and $C$ be a real, symmetric, positive definite $d \times d$-matrix. Does it then hold, that $$ \text{tr}(\Sigma^2C) \geq 0 \quad ?$$
-1
votes
1answer
16 views

Do hermitian matrices commute when they occupy they same elements but have different values?

Given hermitian matrices A and B, they have different values but share the same non zero elements, e.g. $A=\begin{pmatrix}1&0&3\\0&2&4\\3&4&7\end{pmatrix}$ and $B=\begin{...
1
vote
1answer
36 views

Can $\mathfrak{u}(n)$ be decomposed as direct sum of the sets of symmetric and skew-symmetric real matrices?

It is a well-known result (proved for example also in this answer) that $\mathfrak{gl}(n,\mathbb C)\simeq \mathfrak{u}(n)_{\mathbb C}$, which can also be understood as another way to state that any ...
0
votes
1answer
42 views

Inverse of symmetric tridiagonal block toeplitz matrix

There is a triagonal block matrix $M$ of form: $$ M = \begin{bmatrix} A & B^T & 0 & 0 & \cdots & 0 & 0 \\ B & A & B^T & 0 & \cdots & 0 & 0 \\ 0 & B ...
0
votes
0answers
59 views

Signature and Rank of a matrix.

Yes, there are some other answers in regards to signature and rank. However, the definitions my instructor gave are a bit different than what is on WolframAlpha. Also, I thought it would be nice to ...
3
votes
2answers
43 views

How to show that a symmetric matrix that is strictly diagonal dominant (sdd) is is positive and definite?

Let $A\in M_n(\mathbb{R})$ be a real symmetric matrix that is strictly diagonal dominant (sdd) with positive diagonal values, that is, $$\forall i, a_{ii} = |a_{ii}|>\sum_{j\neq i}|a_{ij}|.$$ I ...
2
votes
1answer
80 views

$XA=A^TX$ prove $X$ symmetric matrix

Let $A$ be a nonderogatory matrix. This means the characteristic polynomial and the minimal polynomial of $A$ are coincide. Or, equivalently, every matrix $X$ that satisfies $XA=AX$ can be written as ...
0
votes
1answer
47 views

Explain whether this matrix is symmetric or not?

I have a matrix $M$ and another $N$. $N$ is an orthogonal (orthogonal => $N^{T} = N^{-1})$ r x r matrix and $M$ is an r x r skew symmetric matrix (skew syemmtric => $M^{T} = -M$). Is $(N^{-1})$$(M^2)$$...