Questions tagged [symmetric-matrices]
A symmetric matrix is a square matrix that is equal to its transpose.
3
votes
0answers
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{...
0
votes
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 ...
1
vote
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
vote
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
votes
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 ...
1
vote
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 ...
0
votes
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 $...
0
votes
0answers
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)}}
$$
where $A$ is ...
0
votes
0answers
19 views
Solving a constrained linear system of equations
I want to solve the linear system of equations
$$
\begin{equation}
Ax = b \\
\text{s.t} \quad x \geq 0
\end{equation}
$$
The matrix A is a sparse symmetric matrix.
What is the best time efficient ...
3
votes
2answers
71 views
Show a specially defined matrix is positive definite
Let $E_1, ..., E_n$ be non empty finite sets. Show that the matrix $A = (A_{ij})_{1 \leq i, j \leq n}$ defined by $A_{ij} = \dfrac{|E_i \cap E_j|}{|E_i \cup E_j|}$, is positive semi-definite.
This is ...
0
votes
0answers
26 views
Determinant of a symmetric Hankel matrix with non-zero diagonal elements [on hold]
If $S_r= a^r + b^r + c^r,$ then what is the determinant of the following $3\times 3$ matrix?
\begin{bmatrix}
S_0 & S_1 & S_2\\
S_1 & S_2 & S_3\\
S_2 & S_3 & S_4
\end{bmatrix}
0
votes
2answers
33 views
Symmetrical matrix determinant
Is it true that every symmetrical matrix has a determinant non-zero?
If so how can I prove it?
Note: A symmetrical matrix that is not zero. (thanks to the commenter that pointed it out)
0
votes
2answers
29 views
Sum of symmetric, positive semidefinite matrices
Let $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{p \times n}$. Show that $A^{T}A+ B^{T}B$ is invertible if and only if $\ker A \cap \ker B =\lbrace 0 \rbrace$.
I could show that if it's ...
0
votes
0answers
14 views
Proof check: Sylvester's law of inertia
I was trying to prove the following:
Claim(Sylvester's law of inertia): Let $A$ be a real $n\times n$ symmetric matrix. It is known that there exists an invertible real matrix $P$ such that $$P^tAP=...
4
votes
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
votes
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
votes
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?
1
vote
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
votes
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 ...
2
votes
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
$$M_4=\begin{pmatrix}1 & a & a^2 & a^3 \\ a & 1 & a & a^2 \\ a^2 & a & 1 & a \\ a^3 & a^2 & a & 1\...
0
votes
1answer
37 views
Symmetric complex matrix
Is this matrix symmetric?
$$ A=
\begin{pmatrix}
2 & 0 & (1+i) \\
0 & 3 & -1 \\
(1-i) & -1 & 3 \\
\end{pmatrix}
$$
The $a_{13}$ and $a_{31}$ entries are ...
0
votes
0answers
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 ...
-1
votes
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 ...
-1
votes
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
votes
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)$...
0
votes
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
votes
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$?...
1
vote
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 $...
1
vote
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)$$
It has trace $0$ and thus its eigenvalues are of the form ...
0
votes
0answers
13 views
Constructing Symmetric Hadamard Matrices
I'm trying to understand the exact degrees of freedom involved in constructing Hadamard matrices with elements in $\{1, -1\}$, and if possible, reduce them in such a way that they can be ...
0
votes
2answers
39 views
Orthogonal matrix multiplied by diagonal matrix multiplied by transpose of the orthogonal matrix
Suppose tall matrix $A$ is $n \times k$ and that its columns are orthogonal, i.e., $A' A = I_k$. Suppose further that diagonal $M$ is $n \times n$ and has either $1$ or $0$ on its main diagonal. ...
0
votes
1answer
20 views
Determinant of large square matrix (term by term multiplication with same size matrix)
M is an N by N matrix with coefficients $a_{ij}$, B an N by N matrix with coefficients $b_{ij}$. Both M and B are symmetric matrices. I am trying to write the determinat of a matrix say, M .* B where ...
0
votes
1answer
44 views
The inverse of a submatrix of an inverse matrix; how to numerically approach this?
Suppose that $A$ is a positive-definite symmetric matrix. I want to solve
$$\widetilde{(A^{-1})_k}x = b$$
where $\widetilde{(B)}_k$ is the matrix $B$ but with the $k^{th}$ row and $k^{th}$ column ...
2
votes
1answer
36 views
Result for $A^{-\frac{1}{2}}A(A^{-\frac{1}{2}})^T$
Let $A_n$ be a symmetric square positive definite matrix. (A variance and covariance matrix.)
So, can I say $A^{-\frac{1}{2}}A(A^{-\frac{1}{2}})^T = I_n$?
0
votes
2answers
50 views
Does orthogonal eigenvectors imply symmetric matrix?
If an $n \times n$ matrix $\mathbf A$ is diagonalizable, and orthogonal eigenvectors of $\mathbf A$ form a basis of $R^n$, then is $\mathbf A$ symmetric?
Here is what I tried:
Suppose {${\vec{a_1},\...
1
vote
0answers
32 views
Is this matrix defined from an integral of a non-negative function positive definite?
I have a function $f(x)$ defined as
\begin{equation}
f(x) := \sum_{n = 1}^N c_n f_n(x) = \mathbf{f}(x)^T \mathbf{c},
\end{equation}
where
\begin{align}
\mathbf{f}(x) := \begin{bmatrix} f_1(x) \\ ...
0
votes
1answer
30 views
Prove that $\phi(f(X),Y)=\phi(X,f(Y))~\forall X,Y\in\mathbb R^3$ where $\phi(X,Y)=X^TAY$ and $f:\mathbb R^3\to \mathbb R^3, X\mapsto BX$
Within this AoPS thread there was the following question asked
Let $\phi(X,Y)=X^TAY$ be a scalar product on $\mathbb R^3,$ and let $f:\mathbb R^3\to \mathbb R^3, X\mapsto BX$, where
$$A=\begin{...
0
votes
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\...
1
vote
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: $$...
2
votes
0answers
20 views
Does a square root matrix of a circulant correlation matrix with positive entries also have all positive entries?
I have a circulant correlation matrix that has only positive entries. (Because it is a correlation matrix, it is symmetric with diagonal entries of 1.) I am wondering about the entries of the square ...
1
vote
3answers
40 views
How to show that this matrix is positive semidefinite?
Using the definition, show that the following matrix is positive semidefinite.
$$\begin{pmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 & 15\end{pmatrix}$$
In other words, if ...
0
votes
0answers
17 views
How to generate matrix with non-negative singular values but the matrix is NOT symmetric?
All:
For testing purpose, we would like to generate a sequence of square matrices whose
singular values are non-negative (but real), and the matrices are NOT symmetric.
Is there a simple way to ...
7
votes
1answer
48 views
Show that eigenvalues are symmetric with respect to the origin
The matrices that I am considering are
$$
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 ...
0
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....
