All Questions
744 questions
2
votes
0
answers
79
views
$|Ax|\leq |Bx|$ iff $A\leq B$
Let $A,B$ be positive semidefinite self-adjoint operator on some finite inner product space. Is it true that $|Ax|\leq |Bx|$ for every vector $x$ iff $A\leq B$ [this notation means $B-A$ is positive ...
- 1,194
1
vote
1
answer
47
views
Given a positive definite matrix S, and a symmetric matrix A with small errors as S. How to prove that A is positive definite?
This is problem 4.14 in Multivariable Calculus with Applications by Peter D. Lax & Maria Shea Terrell.
A symmetric 2 by 2 matrix $A$ has been computed numerically with small errors as a symmetric ...
- 13
1
vote
1
answer
69
views
Prove a lower bound on eigenvalues of a class of matrices
I am not sure if this is trivial or not. Let $\mathbf{A} \in \mathbb{R}^{n \times n}$ be real and symmetric, with the property that
$$
\mathbf{A}_{ii} = 1, \quad \mathbf{A}_{ij} \in [0,1], \quad i,j \...
- 514
0
votes
0
answers
28
views
Is there an explicit formula for the eigenvalues and eigenvectors of this particular symmetric matrix? [duplicate]
Consider a symmetric $n\times n$ matrix with the elements $A_{ij}=\min(i,j)$, i.e.:
$$A=\begin{pmatrix}
1 & 1 & \cdots & 1 \\
1 & 2 & \cdots & 2 \\
\vdots & \vdots & \...
- 338
0
votes
1
answer
40
views
Semi-definiteness of symmetric matrix with some zero diagonal elements
Let we have a symmetric matrix
$$
A=A^T\in\mathbb R^{n\times n}
$$such that some of its diagonal elements are zero.
Is it true that $A$ is semidefinite only if it is diagonal?
- 51
2
votes
0
answers
51
views
Lower Bound on the Minimum Eigenvalue of a Hollow Symmetric Matrix with Entries in $[0,1]$
Let $\mathbf{A}$ be a hollow symmetric real $n \times n$ matrix, where each entry $A_{i,j} \in [0,1]$ and $A_{i,i} = 0$ for all $i$. I would like to establish a lower bound for the minimum eigenvalue $...
- 51
8
votes
0
answers
287
views
Bounds on the Eigenvalues of Perturbations of a Symmetric Matrix
Let us fix $\varepsilon\in (0,1)$ and $\beta\in\mathbb R$. Consider the $2 n\times 2n$ symmetric tridiagonal probability matrix
$$Q_n :=\begin{bmatrix}
1-\frac{ε}{2} & \frac{ε}{2} & & &...
- 3,176
1
vote
0
answers
43
views
A relation about norms of a block matrix.
Suppose $M =
\begin{bmatrix}
A & B \\
B^{T} & C \\
\end{bmatrix}$,
where $A$ and $C$ are $n\times n$ real symmetric matrices and $B$ is an arbitrary $n\times n$ real matrix. By calculating ...
- 127
1
vote
1
answer
62
views
How to recover the vector $x$ when multiplying it with a matrix $A$ that has a specific structure
I have the matrix with specific structure as below:
$$A = \left[ \begin{matrix}
a & -b& -c& d \\
b & e& -d& -f \\
c & -d& a& -b \\
d & f& b & ...
- 167
3
votes
1
answer
83
views
For what $A$ the linear map $f(B):=AB^T+BA^T$ surjective, defined as a linear map between suitable vector spaces defined below?
Let $V$ denote the space of $m\times m$ real square matrices. Let $W$ denote the space of $m\times m$ real symmetric square matrices. For an $m\times m$ square (not necessarily symmetric) matrix $A,$ ...
- 2,556
0
votes
1
answer
78
views
Rank of matrix $a_i \cdot b_j + a_j \cdot b_i$ [duplicate]
Find the rank of the matrix $d_{ij} = a_i \cdot b_j + a_j \cdot b_i$, where $a$ and $b$ are arbitrary vectors.
I noticed that
$C = a \cdot b^T = \begin{bmatrix}
a_{1} \\
... \\
...
- 371
1
vote
1
answer
51
views
Suppose that the eigenvalue of largest magnetude of $P$ is simple. Is it true that the the same happens to $U P$ for any unitary matrix $U$?
Suppose that $P = [p_{i,j}]_{1\leq i,j\leq n}$ is an $n\times n$ matrix such that:
$p_{i,j}\geq 0$ for all $i,j\in\{0,1,\ldots,n\};$
$\sum_{j=1}^n p_{i,j}=1$;
$P$ is a symmetric matrix, $p_{i,j}=p_{j,...
- 3,176
2
votes
1
answer
58
views
Rank of a matrix $A$
Let $$A=\begin{pmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{13} & a_{14} & a_{24} \\
a_{13} & a_{14} & a_{24} & a_{34} \\
a_{14} & a_{24} & a_{...
- 646
1
vote
2
answers
68
views
Congruent diagonalization using row and column operations
Let $$A=\begin{pmatrix}
1 & 2 & 3\\
2 & 4 & 6\\
3 & 6 & 9 \end{pmatrix}.$$
Find an invertible matrix $P$ such that $P^tAP$ is diagonal.
Let me start by saying that I already ...
- 1,304
0
votes
1
answer
39
views
Is it true that $D A P D^T = A D P D^T$ if $P$ is symmetrical, positive definite and $D$ is diagonal?
I know in general, matrix multiplication is not commutative, but would it be true in this special case?
$D A P D^T = A D P D^T$ where $A, D, P$ are all $n by n$ matrix. But $P$ is symmetrical and ...
- 199
1
vote
0
answers
50
views
How to decompose the Hessian matrix of a 3rd degree polynomial into 2 or more vectors/matrices?
I am trying to figure out what can be said about the spectral radius of the Hessian of a 3rd degree random polynomial defined over a unit hypercube, drawing on known results in random matrix theory. ...
- 113
1
vote
0
answers
21
views
Fastest way to divide by a symmetric positive matrix
Say $P_{yx}$ is a general $(n_y,n_x)$ matrix.
Say $P_y$ is symmetric, positive definite, of size $(n_y,n_y)$.
I want to compute $GT=P_y\backslash P_{yx}$ (matrix left division, or perhaps more ...
- 280
1
vote
1
answer
40
views
A matrix inequality $X - \frac{1}{u^\top Xu} Xuu^\top X \succeq Y - \frac{1}{u^\top Yu} Yuu^\top Y$
Problem. Let $n\ge 2$. Let $X, Y$ be $n\times n$ real symmetric positive definite (PD) matrices with $X \succeq Y$ (i.e. $X - Y$ is positive semi-definite (PSD)). Let $u\ne 0$ be a $n\times 1$ real ...
- 42.7k
1
vote
0
answers
75
views
A question about permutation similarity of symmetric matrices
Let $A , B \in M_{4 \times 4}(\mathbb{R}_{\geq 0})$. Matrices $A, B$ are $\textbf{permutation similar}$ if there exists a permutation matrix such that $A= PBP^T$. Define $\mathrm{diag}(A)$ to be the ...
0
votes
0
answers
39
views
Equivalence for a matrix to be symmetric
Let $A\in\mathbb R^{n\times n}$ with the QR-decomposition $A=QR$ ($Q$ orthogonal, $R$ upper triangular matrix). I asked myself if the following statement holds:
$$A\in\mathbb R^{n\times n}\text{ ...
- 21
0
votes
0
answers
48
views
Inverse and Determinant of Matrix $Axx^TA+cA$
Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
- 2,953
1
vote
0
answers
36
views
Singular values as min max of absolute rayleigh quotient
Consider a real symmetric matrix M, satisfying $\mathbf{1}^\intercal M = \mathbf{1}^\intercal$ having eigenvalues $1=\lambda_1 \gt\|\lambda_2\| \geq \|\lambda_3\| .... \geq \| \lambda_n\|$, then can I ...
0
votes
1
answer
31
views
Lyapunov Special Symmetric Case [closed]
Consider the Lyapunov equation: $$AX+XA = B$$ and assume that $A$ is symmetric positive definite and $B$ is symmetric.
I am not able to proof that $X=X^{T}$ holds.
Would be grateful if somebody could ...
- 19
0
votes
2
answers
54
views
Writing convention of Courant–Fischer theorem
Let $A \in \mathcal{M}_n(\mathbb{R})$ be a symmetric matrix and $\lambda_1 \leq \lambda_2\dots\leq\lambda_n$ be its real eigenvalues taken with multiplicities. Let $1\leq i_1 \leq i_2\dots\leq i_k\leq ...
user1290851
0
votes
2
answers
89
views
Why $X^\perp=\text{span}\space\{x_{k+1}\dots,x_n\}$ true?
Given the set $X=\{x_1,x_2,\dots\,x_k\}$ be a orthonormal set of eigenvectors of a symmetric matrix $A\in\mathcal M_n(\mathbb{R})$. Then I don't understand why $X^\perp=\text{span}\space\{x_{k+1}\dots,...
user1290851
0
votes
1
answer
310
views
Different versions of Courant–Fischer theorem
Let $A \in \mathcal{M}_n(\mathbb{R})$ be a symmetric matrix and $\lambda_1 \leq \lambda_2\dots\leq\lambda_n$ be its real eigenvalues taken with multiplicities. Let $1\leq i_1 \leq i_2\dots\leq i_k\leq ...
user1290851
1
vote
1
answer
98
views
Prove that if $\lambda$ is an eigenvalue of a symplectic matrix, then $\frac{1}{\lambda}$ is also an eigenvalue of such matrix
I´m trying to solve the following problem:
A symplectic $n\times n$ matrix $A$ follows this conditions:
$J$ is a $n\times n$ matrix
$J^2=-I$
$A^TJA=J$
$n$ is an even number
Prove that if $\lambda$ ...
- 187
5
votes
1
answer
155
views
Solving Xa=b for an unknown matrix X
I'm interested in studying the solutions of $Xa=b$ for an unknown square matrix $X$, and given (known) column vectors $a$ and $b$ in $\mathbb{R}^n$.
For any numerical $a, b$, one can directly attempt ...
- 467
0
votes
0
answers
50
views
Given a symmetric diagonally dominant matrix, does $-|a_{ii}|\le \sum_{j\neq i} a_{ij}$ hold?
I am studying linear algebra and I am playing with some basic notion (symmetric matrices, pivots, diagonal matrices etc).
Let $A\in\mathbb R^{n\times n}$ be a symmetric matrix and consider the ...
- 425
0
votes
0
answers
78
views
There is a permutation matrix $P$ such that $PAP^{T}$ is in this form for symmetric $A$
Suppose that $A$ is a real matrix and is symmetric and nonzero, then I want to prove that there is a permutation matrix $P$ such that $PAP^{T}=\left[\begin{array}{ll}
B & E^{\top} \\
E & C
\...
- 623
2
votes
0
answers
50
views
Counting number of free parameters from a matrix constraint equation
How many free parameters does a $28 \times 28$ symmetric matrix $M$ have if it is subjected to a constraint
$$M^T L M=L \tag{1}$$ where
$$L=\begin{bmatrix}
0&I_6&0\\I_6&0&0\\0&0&...
- 396
-1
votes
1
answer
135
views
Relationship between eigenvalues when the same change is applied to two different adjacency matrices of graphs with the same eigenvalues
I'm attacking to solve the graph isomorphism problem using the adjacent matrices.
I would appreciate it if someone could show me whether following conjecture works or not. I expect this conjecture to ...
0
votes
1
answer
84
views
Is there a symmetric matrix with integer coefficients of order n?
Let $n$ be a positive integer. It is easy to find a symmetric matrix with complex coefficients with order $n$ i.e. $A^n = I$, one can just consider the diagonal matrix with $n$-th roots of units.
A ...
21
votes
1
answer
2k
views
Every matrix is a product of two symmetric matrices
Let $\mathbb{F}$ be a field with char $\mathbb{F} \neq 2$.
Let $A \in M_n(\mathbb{F})$.
Does there exist symmetric matrices $B,C \in M_n(\mathbb{F})$ such that $A=BC$?
The answer is yes when $\mathbb{...
- 321
0
votes
3
answers
117
views
Symmetric property of symmetric matrices [closed]
Are symmetric matrices symmetric about a particular axis geometrically ? Or is the property defined only for the matrix representation of the linear transformation?
If you consider a unit square, ...
- 53
0
votes
0
answers
20
views
Eigendecomposition of a real symmetric matrix obtained by "projection"
I have a real square matrix $s\times s$ that writes
$$ M = PU\Lambda U^\top P^\top $$
with $P = \left( \mathrm{I}_s ,0\right)\in \mathbb{R}^{s\times r}$, $U \in \mathbb{R}^{r\times r}$ orthogonal, and ...
- 1
0
votes
1
answer
68
views
Under which conditions is the sum of two symmetric and invertible matrices invertible?
I am using a numerical method that involves a matrix
$$
Z = X + Y,
$$
where $X$ and $Y$ are both symmetric and invertible. The numerical method then proceeds to invert $Z$ for computations.
In ...
- 477
0
votes
0
answers
29
views
Criterion for a non-trivial overlap of null spaces of real-symmetric matrices
I have two large real-symmetric matrices $A,B\in \mathbb{R}^{N\times N}$, which obey $A \cdot B = 0 = B \cdot A$, meaning that they both commute (and thus share eigenvectors) and anticommute.
I am ...
- 586
0
votes
0
answers
25
views
Matrix-free definition for an operation on symmetric positive semidefinite bilinear forms
Let $SPD_2(\mathbb{R}^n)$ be the set of all symmetric and positive semidefinite bilinear forms $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$. Define a binary operation $(f,g) \mapsto f \ast g$ ...
- 513
0
votes
0
answers
147
views
The matrix exponential of a positive definite matrix is also positive definite [duplicate]
Let $L$ be a $n \times n$ symmetric positive definite matrix. Prove that for $t>0$ that the matrix exponential $e^{-tL}$ is also symmetric positive definite.
- 57
2
votes
0
answers
82
views
When do the eigenvalues of a symmetric matrix belong to its field?
Every polynomial is the characteristic polynomial of a matrix (see companion matrix). Therefore, the eigenvalues of a matrix over a field $\mathbb F$ must belong to $\mathbb F$ if, and only if, $\...
- 5,643
5
votes
1
answer
133
views
When is an invertible symmetric logic matrix unimodular?
I am currently interested in invertible symmetric logical matrices, or, $(0,1)$-matrices, i.e., $n \times n$ matrices whose entries are either $0$ or $1$ (integers). I noticed that many invertible ...
- 293
0
votes
1
answer
82
views
How to prove the inequality $\det\left(\sum_{i=1}^k{A_i^TA_i}\right) \geq 0$ for matrices $A_1, \dots, A_k\in M_n(\mathbb{R})$? [duplicate]
I am exploring a mathematical problem involving two natural numbers, $n$ and $k$ ($n, k \geq 1$), and a collection of matrices $A_1, \dots, A_k \in M_n(\mathbb{R})$. The goal is to establish the ...
user1265622
0
votes
1
answer
68
views
Minimizing $x^T A x$ subject to $B x \leq b$
Given $$ A = \begin{bmatrix} 4 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 1 & 2 & -3 \\ 5 & 2 & 1 \end{bmatrix}, \qquad b = \...
0
votes
1
answer
45
views
For a polar decompoistion $A=BJ$, the matrices $A$ and $B$ commute
Suppose $A$ is an invertible real $n\times n$ matrix, and consider its (unique) polar decomposition $A=BJ$ where $B$ is positive definite symmetric and $J$ is orthogonal. Is it true that $AB=BA$?
...
- 1,920
1
vote
1
answer
147
views
Dimension of $\mathfrak{sp}(2n, \mathbb{R})$
I'm computing the dimension of $\mathrm{Sp}(2n, \mathbb{R})$ and I've shown that its Lie algebra is $\mathfrak{sp}(2n, \mathbb{R})=\{X\in\mathrm{Mat}_{2n}(\mathbb{R})\mid XJ+JX^\mathrm{T}=0\}$, where
$...
- 534
0
votes
1
answer
49
views
Finding the Vector $v$ for a Given Householder Matrix Transformation of Non-Collinear Vectors $a$ and $b$
Consider a vector $v$ in $\mathbb{R}^{n\times1}$. The Householder matrix is defined as follows:
$$H(v)=I-\dfrac{2vv^T}{v^Tv}.$$
It can be demonstrated that $H(v)$ is symmetric and orthonormal. The ...
user1253495
1
vote
1
answer
64
views
Dimension of a Matrix subspace
What is the dimension and the number of basis vectors for a subspace of 3×3 symmetric matrices?
Earlier my professor told us that the dimension and the number of basis vectors for a subspace are the ...
- 13
1
vote
0
answers
22
views
Ordering positive definite matrices with diagonal matrices
Let $A \succ 0$ be a Hermitian positive definite matrix. I'm trying to understand the claim that there exists a (diagonal) positive definite matrix $B \succ 0$ such that $A \prec B$, that is, $B-A$ is ...
- 57
1
vote
0
answers
31
views
Name for 3 full diagonal matrix
I was wondering if there is a name or a terminology for a matrix with 3 full diagonals. It is similar with tridiagonal matrices, where the later is only centered at diagonal.
For such a matrix,
$$\...
- 337