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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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24 views

Show that the spectral norm of one matrix is smaller than the other.

Given matrices $$A = \begin{bmatrix} 0 & 0 & 0 & 1/3 \\ 0 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 1 \\ 1/3 & 1/2 & 1 & 0 \end{...
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0answers
8 views

Relationship between Cayley transform and polar decomposition

I want to (as) efficiently (as possible) numerically compute the rotation $\mathrm{R}$ in the polar decomposition of a $n\times n$ matrix of the form $\mathrm{I} + \mathrm{W}$ where $\mathrm{I}$ is ...
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2answers
40 views

Proving the difference of two matrices is PSD

Claim: For $x\in \mathbb{R}^n$, we have $\operatorname{Diag}(x) - xx^T \succeq 0$ if and only if $x_i \geq 0 \ \forall i\in [n]$ and $\sum_{i} x_i \leq 1$. Where $\operatorname{Diag}$ denotes the ...
1
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3answers
45 views

Finding other eigenvector and matrix $A$ given eigenvalues

I want to find a symmetric matrix $A$, whose eigenvalues are $4$ and $-1$. One of the eigenvectors corresponding to the eigenvalue $4$ is $(2,3)$. I want to find an eigenvector corresponding to the ...
0
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1answer
57 views

When is $\|\boldsymbol A\| \|\boldsymbol A^{-1}\|$ bounded?

According to the sub-multiplicative property of (some) matrix norms, we know that \begin{equation} \| \boldsymbol I \| \leq \|\boldsymbol A\| \|\boldsymbol A^{-1}\| \ , \end{equation} for some ...
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2answers
37 views

If both $A-B$ and $B-A$ are positive semidefinite, then $A = B$

Let $A, B$ be two positive semidefinite matrices. Prove that if both $A-B$ and $B-A$ are positive semidefinite, then $A = B$. I can show that their diagonal elements are the same but for others I ...
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0answers
12 views

Williamson's theorem for positive semi-definite matrices of even size

I know the Williamson's theorem for positive definite matrices of even size. I was wondering if the theorem also holds for positive semi-definite matrices with even rank. More precisely, if $A$ is a $...
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14 views

Pseudo determinant of a non negative matrix (singular matrix here ).

This wikipedia article gives the following formula for finding the pseudo determinant of a matrix. $$ |A|_+ = \lim\limits_{\alpha\to0} \frac{|A+\alpha I|}{\alpha^{n-\mathrm{rank}(A)}} $$ where $A$ is ...
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0answers
19 views

Solving a constrained linear system of equations

I want to solve the linear system of equations $$ \begin{equation} Ax = b \\ \text{s.t} \quad x \geq 0 \end{equation} $$ The matrix A is a sparse symmetric matrix. What is the best time efficient ...
3
votes
2answers
71 views

Show a specially defined matrix is positive definite

Let $E_1, ..., E_n$ be non empty finite sets. Show that the matrix $A = (A_{ij})_{1 \leq i, j \leq n}$ defined by $A_{ij} = \dfrac{|E_i \cap E_j|}{|E_i \cup E_j|}$, is positive semi-definite. This is ...
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26 views

Determinant of a symmetric Hankel matrix with non-zero diagonal elements [on hold]

If $S_r= a^r + b^r + c^r,$ then what is the determinant of the following $3\times 3$ matrix? \begin{bmatrix} S_0 & S_1 & S_2\\ S_1 & S_2 & S_3\\ S_2 & S_3 & S_4 \end{bmatrix}
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2answers
33 views

Symmetrical matrix determinant

Is it true that every symmetrical matrix has a determinant non-zero? If so how can I prove it? Note: A symmetrical matrix that is not zero. (thanks to the commenter that pointed it out)
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2answers
29 views

Sum of symmetric, positive semidefinite matrices

Let $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{p \times n}$. Show that $A^{T}A+ B^{T}B$ is invertible if and only if $\ker A \cap \ker B =\lbrace 0 \rbrace$. I could show that if it's ...
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0answers
14 views

Proof check: Sylvester's law of inertia

I was trying to prove the following: Claim(Sylvester's law of inertia): Let $A$ be a real $n\times n$ symmetric matrix. It is known that there exists an invertible real matrix $P$ such that $$P^tAP=...
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3answers
124 views

How to solve $A^{\frac 12} B A^{\frac 12} = C$ for $A$?

Suppose that matrices $A,B,C$ are symmetric and positive definite. Then, $A$ has a unique, positive square root, which we call $A^{\frac 12}$. If $$A^{\frac 12} B A^{\frac 12} = C$$ then can we write ...
0
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1answer
28 views

How to prove the transpose matrix is in a vector space with restrictions on the dimension

For an assignment in class, I have the following question. Let n $\geq 1$ and let W be a subspace of $Mn\times n(K)$ such that $dim(W)>\frac{n^2-n}{2}$. Prove that W contains a non-zero matrix ...
0
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1answer
45 views

Show that A is skew-symmetric if and only if $x^tAx = 0$

I've tried by starting with setting $x^tAx = 0 = x^t(-A^t)x$ and checking it termwise, but I don't think this will show me anything. Could you explain how to approach this problem please?
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0answers
21 views

Inverse of a matrix which is difference of a singular matrix with a small diagonal matrix?

If $A$ is a real symmetric singular matrix (similar to a Laplacian matrix, which comes from M'GM, where M is incidence matrix and G is a diagonal matrix) with large values and $B$ is a diagonal matrix ...
0
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0answers
17 views

Cholesky factor when adding a row and column in between

I have a problem where I have the Cholesky factorization ($A=LL'$) of a symmetric positive-definite matrix. Now, I need to add a new row and column somewhere in the "middle" of the matrix and compute ...
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0answers
27 views

Looking for properties of, or formulae for eigenvalues of a symmetric matrix reminiscent of Toeplitz matrices

I'm looking at $N\times N$ matrices $M_N$ of the form $$M_4=\begin{pmatrix}1 & a & a^2 & a^3 \\ a & 1 & a & a^2 \\ a^2 & a & 1 & a \\ a^3 & a^2 & a & 1\...
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1answer
37 views

Symmetric complex matrix

Is this matrix symmetric? $$ A= \begin{pmatrix} 2 & 0 & (1+i) \\ 0 & 3 & -1 \\ (1-i) & -1 & 3 \\ \end{pmatrix} $$ The $a_{13}$ and $a_{31}$ entries are ...
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12 views

Which type of factorization is appropriate?

I have a symmetric matrix (square)($A$) with positive values and zero on its main diagonal. I need to find a matrix $Y$ which is: $Y^TY = A$ I don't have any non-negativity constraint on the ...
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1answer
88 views

$\log(\det A) = tr(\log(A))$? [duplicate]

I want to know that if the equation in the title always holds? I have generated a random Hermitian matrix $A$, and then compute $\log(\det A)$ and $ tr(\log(A))$ in matlab, it is not equal. So I'm ...
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0answers
16 views

What is the significance of the max and min values of a inverted matrix?

so I have a lab where you have to find the inverse of a symmetrical 16 x 16 matrix, find the maximum and minimum value of the inverse using MATLAB, and describe the significance of those extreme ...
0
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1answer
45 views

Matrix norm of $A-B$ and their smallest eigenvalues

Let $A$ and $B$ be symmetric positive definite matrices of size $n\times n$ such that $\|A - B\| < \frac{1}{4}\lambda_{\min}(B)$ where $\|\cdot\|$ is the matrix 2-norm and $0 < \lambda_{\min}(B)$...
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0answers
13 views

Conditions for the unique symmetric matrix decomposition

Let $V=CC'$ ($dim(C)=(n,k),k<n$) be a symmetric $(n,n)$ real matrix produced from unknown full-rank matrices C. I want to decompose it uniquely into the product $AA'$, $dim(A)=(n,k)$. What ...
0
votes
1answer
15 views

Prove the inverse of a nonnegative matrix is nonnegative

Defintion of a nonnegative matrix: Symmetrical matrix $A: n \times n$ is non-negatively defined when $A > 0$ or $A ≥ 0$ We have to prove the following: If $A$ is defined as a nonnegative matrix, ...
0
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3answers
35 views

Find 3x3 matrix by determinant and 2 eigenvalues/-vectors

I have two eigenvectors: $(2, 1, -1)'$ with eigenvalue $1$, and $(0, 1, 1)'$ with eigenvalue $2$. The corresponding determinant is $8$. How can I calculate the $3\times3$ symmetric matrix $A$ and $AP$?...
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2answers
77 views

can any symmetrix matrix be written as $A^TA$?

Let $A$ be a $m\times n$ real matrix. Then $B:=A^TA$ is an $n\times n$ symmetric matrix. Is the converse true? More precisely, given any $n\times n$ symmetric matrix $B$ and any fix positive integer $...
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0answers
22 views

What is the most general matrix that gives rise to all even characteristic polynomials?

Is there some general form of all matrices which give rise to all even or all odd characteristic polynomial terms? For skew-symmetric matrices such that $A^T=-A$ we necessarily have all even or all ...
3
votes
1answer
92 views

What type of matrices lead to $\pm$ eigenvalues?

I have a code which is spitting out matrices of the form $$\left(\matrix{0&a&-a\\a&+\gamma&0\\-a&0&-\gamma}\right)$$ It has trace $0$ and thus its eigenvalues are of the form ...
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0answers
13 views

Constructing Symmetric Hadamard Matrices

I'm trying to understand the exact degrees of freedom involved in constructing Hadamard matrices with elements in $\{1, -1\}$, and if possible, reduce them in such a way that they can be ...
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2answers
39 views

Orthogonal matrix multiplied by diagonal matrix multiplied by transpose of the orthogonal matrix

Suppose tall matrix $A$ is $n \times k$ and that its columns are orthogonal, i.e., $A' A = I_k$. Suppose further that diagonal $M$ is $n \times n$ and has either $1$ or $0$ on its main diagonal. ...
0
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1answer
20 views

Determinant of large square matrix (term by term multiplication with same size matrix)

M is an N by N matrix with coefficients $a_{ij}$, B an N by N matrix with coefficients $b_{ij}$. Both M and B are symmetric matrices. I am trying to write the determinat of a matrix say, M .* B where ...
0
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1answer
44 views

The inverse of a submatrix of an inverse matrix; how to numerically approach this?

Suppose that $A$ is a positive-definite symmetric matrix. I want to solve $$\widetilde{(A^{-1})_k}x = b$$ where $\widetilde{(B)}_k$ is the matrix $B$ but with the $k^{th}$ row and $k^{th}$ column ...
2
votes
1answer
36 views

Result for $A^{-\frac{1}{2}}A(A^{-\frac{1}{2}})^T$

Let $A_n$ be a symmetric square positive definite matrix. (A variance and covariance matrix.) So, can I say $A^{-\frac{1}{2}}A(A^{-\frac{1}{2}})^T = I_n$?
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2answers
50 views

Does orthogonal eigenvectors imply symmetric matrix?

If an $n \times n$ matrix $\mathbf A$ is diagonalizable, and orthogonal eigenvectors of $\mathbf A$ form a basis of $R^n$, then is $\mathbf A$ symmetric? Here is what I tried: Suppose {${\vec{a_1},\...
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0answers
32 views

Is this matrix defined from an integral of a non-negative function positive definite?

I have a function $f(x)$ defined as \begin{equation} f(x) := \sum_{n = 1}^N c_n f_n(x) = \mathbf{f}(x)^T \mathbf{c}, \end{equation} where \begin{align} \mathbf{f}(x) := \begin{bmatrix} f_1(x) \\ ...
0
votes
1answer
30 views

Prove that $\phi(f(X),Y)=\phi(X,f(Y))~\forall X,Y\in\mathbb R^3$ where $\phi(X,Y)=X^TAY$ and $f:\mathbb R^3\to \mathbb R^3, X\mapsto BX$

Within this AoPS thread there was the following question asked Let $\phi(X,Y)=X^TAY$ be a scalar product on $\mathbb R^3,$ and let $f:\mathbb R^3\to \mathbb R^3, X\mapsto BX$, where $$A=\begin{...
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2answers
32 views

how prove that a linear transformation is diagonalizable, given an eigenvalue and the dimension of its kernel

A question from an exam : (First year mechanical engineering, first course in linear algebra): Let $V$ be the vector space of $2\times2$ matrices, and let $U$ be the subspace of $V$ containing $2\...
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0answers
49 views

Linear algebra- Diagonalization of a symmetric matrix

A linear transformation $$ T:R^3→R^3$$ is defined as $$ T:(x)=Cx$$ where $C$ is a symmetric matrix. a) State the dimensions of the eigenspaces $\mbox{N(C-αI)}$ and $N(c-βI)$ It is also given that: $$...
2
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0answers
20 views

Does a square root matrix of a circulant correlation matrix with positive entries also have all positive entries?

I have a circulant correlation matrix that has only positive entries. (Because it is a correlation matrix, it is symmetric with diagonal entries of 1.) I am wondering about the entries of the square ...
1
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3answers
40 views

How to show that this matrix is positive semidefinite?

Using the definition, show that the following matrix is positive semidefinite. $$\begin{pmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 & 15\end{pmatrix}$$ In other words, if ...
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0answers
17 views

How to generate matrix with non-negative singular values but the matrix is NOT symmetric?

All: For testing purpose, we would like to generate a sequence of square matrices whose singular values are non-negative (but real), and the matrices are NOT symmetric. Is there a simple way to ...
7
votes
1answer
48 views

Show that eigenvalues are symmetric with respect to the origin

The matrices that I am considering are $$ M = \begin{bmatrix} A & B \\ C & -A^\top \end{bmatrix}, $$ with $A,B,C\in\mathbb{C}^{n\times n}$, $C = C^\top$ and $B = B^\top$. I noticed from ...
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votes
2answers
27 views

Symmetric matrices vector space property

Let define $A=\{X\in\mathbb{C}^{3,3}:X=X^T\}$ where $U^T$ is a transpose of the matrix $U$. Then the $A$ is a vector space of symmetric matrices. Now how can we prove that $\{Y\in\mathbb{C}^{3,3}:\...
1
vote
1answer
30 views

How to prove that $\langle P,A^2 \rangle \le 0 $ for every positive $P$ and skew-symmetric $A$?

I have stumbled upon the following claim, and I wonder if it has a simple proof: Let $P$ be a real $n \times n$ symmetric positive definite matrix. Then for every real skew-symmetric matrix $A$, $\...
1
vote
1answer
43 views

Symmetric integer matrix and odd entry in the diagonal.

K is a 2N$\times$2N symmetric integer matrix with at least one odd element in the diagonal. Suppose $\mathcal{M}$ be a set of integer vectors satisfying the following two properties: 1) $m^{T}K^{-1}m'...
6
votes
0answers
63 views

Determinant of a symmetric matrix with entries on diagonals

I am interested in the calculation of the determinant of the $N\times N$ symmetric matrix \begin{equation*} \mathbf B = \left(\begin{array}{*{20}c} 2 & & -1& &-1& &\\ & 2 &...
0
votes
1answer
38 views

Find A a 3x3 real symmetric matrix [closed]

Find A a 3x3 real symmetric matrix such that the linear map A: R3 to R3 with x to Ax satisfies null(A) = {a(1,1,1) : a in R} If anyone has a slow detailed explanation, I'd appreciate it! https://i....