All Questions
Tagged with linear-algebra matrix-analysis
364 questions
8
votes
1
answer
490
views
Determinants (and traces) of linear maps of matrices
Let $k$ be a field or a commutative ring with unit and let $F:M_n(k)\to M_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A_1XB_1 + \cdots + A_m X B_m$ for some $A_i,B_i\in ...
- 7,127
1
vote
1
answer
437
views
Eigenvector of a nonnegative matrix in closed form
Consider $n\times 1$ vector $\alpha = (\alpha_{1}, ..., \alpha_{n})$, where $0<\alpha_{i}<1$, and $\sum_{i=1}^{n}\alpha_i = 1$. Construct the $n\times n$ zero-diagonal matrix $A$ with $(i,j)$-th ...
- 1,538
0
votes
2
answers
160
views
A matrix between vectors, and inequality!
I have an inequality as follows
$$s^T\phi\leq -|s|^TA$$
where $s$, $\phi$ and $A$ are vectors with appropriate dimensions. I want to prove that this inequality holds for the following too
$$s^TM\...
- 29
1
vote
1
answer
622
views
Can I modify the singular values of a matrix in order to get a negative eigenvalue?
Let $A \in \mathbb{R}^{n \times n}$ be a real nonsymmetric matrix with eigenvalues $\left\{\lambda_i : i=1..n\right\}$ with positive real part $\Re(\lambda_i) > 0$ $\forall i=1..n$
Let $A=U\Sigma ...
- 323
1
vote
0
answers
76
views
When is $F(X)BF(X)$ operator monotone, if $F(X)$ is operator monotone?
Let $\Omega_{n}$ denote the cone of $n\times n$ real symmetric positive definite matrices, and consider $F:\Omega_{n} \mapsto \Omega_{n}$. For $X,Y \in \Omega_{n}$, the matrix valued function $F(\cdot)...
- 1,538
4
votes
2
answers
1k
views
Derivative of pseudoinverse with respect to original matrix
I have been trying to find an analytical expression for the following:
$\frac{\partial {X^{+}}}{\partial {X}}$
In my case, $X$ has a constant rank.
I've found the formula for differentiating a ...
- 143
0
votes
1
answer
61
views
Define a matrix function with a specific property
Let $S$ be the set of all positive semidefinite Hermitian matrices of order $mn$ over $\mathbb{C}$. Any matrix $H$ can be partitioned into blocks $H_{ij}$ of order $n$ that is $H_{mn \times mn} = (H_{...
- 363
1
vote
1
answer
82
views
Why $ \langle \mathcal{R}_{\Omega}\,X\,,\mathcal{R}_{\Omega}\,X\rangle \ge \langle X\,,\mathcal{R}_{\Omega}\,X\rangle $?
Let $\{w_{a}\}_{a=1}^{n^2}$ be an orthonormal basis for a matrix. Assume we pick $m$ elements of this basis uniformly at random. $\Omega$ denotes the set of randomly chosen basis. With that, in matrix ...
- 221
3
votes
0
answers
298
views
Singular value decomposition of a low rank weak diagonally dominant M-matrix. When is the unitary polar matrix positive semi-definite?
Let $A$ be an $n \times n$, non-symmetric, real, weak diagonally dominant M-Matrix. Its diagonal is strictly positive, its off-diagonal is negative or zero and all its columns sum to zero. $A$ has ...
- 323
1
vote
0
answers
187
views
A question concerning positive definite matrix functions
Let $C(e^{i\theta})$ be an $m\times n$ ($m\ge n$) matrix-valued continuous function of $\theta\in[-\pi,\pi]$. Let $A_1(e^{i\theta})$ and $A_2(e^{i\theta})$ be two $n\times n$ positive definite matrix-...
- 2,712
4
votes
2
answers
311
views
Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$
For given $n\times n$ real symmetric positive definite matrices $X_{1}, X_{2}$, let $Y := X_{1}^{-1}X_{2}$, and let $I$ be the identity matrix.
I would like to solve the following equation for the ...
- 1,538
5
votes
1
answer
2k
views
Diagonalization of real symmetric matrices with symplectic matrices
Consider the following real symmetric matrix
$M=\left[\begin{array}{ccc} A & B\\ B^T & D \end{array}\right]$
Both A and D are real symmetric $n\times n$ matrices. B is a real $n\times n$...
- 51
1
vote
1
answer
1k
views
product of Gaussian random matrix and a deterministic diagonal matrix
Suppose that $G$ is an $n\times n$ Gaussian random matrix of i.i.d. entries $N(0,1/n)$ and $D$ is an $n\times n$ deterministic diagonal elements. I'd like to know if there have been results on the ...
- 640
2
votes
1
answer
3k
views
The $n$th power of a matrix by Companion matrix
At first, I want to explain why did I say the $n$th power of a matrix by companion matrix. Suppose that $A$ is a matrix
of order $d$ over an ordinary field. There are several methods to obtain a ...
- 313
4
votes
3
answers
3k
views
Is this inequality involving the Frobenius norm right?
Let $A$ be a generic (or varying) square, real $ n \times n$ matrix. Let $G$ be a fixed $n \times k$ matrix, $k < n.$ Denote by $||.||_F$ the Frobenius norm.
Is it true that $||AG||_F \geq c(G) ||...
- 1,467
2
votes
0
answers
147
views
Is the following inequality true for the norm of Moore-Penrose pseudoinverses?
Let $L$ be a real, positive semi-definite, symmetric, square matrix, with pseudoinverse $L^{+}$. It can be shown for the operator norms $||.||_{op}$ that: if $L$ is invertible and $||I - L||_{op} < ...
- 1,467
0
votes
1
answer
350
views
The order of companion matrix over various modulo
We consider a positive integer number and call it our modulo and denote it with $m$. We choose a positive integer number like $p$ and
call it the degree of our polynomial. We select $p$ integer ...
- 313
6
votes
1
answer
882
views
A question on the smallest singular value
Let $X(r)$ be the set of matrices $A \in M(n \times m)$, $n \leq m$, such that the norm of $A$ (largest singular value) is smaller or equal than $1$ and the smallest singular value of $A$ is smaller ...
- 61
9
votes
1
answer
492
views
When are two binary matrices simultaneously equivalent to their transpose?
For any real square matrix $A$ there is an invertible matrix $P$ such that $A^t = P^{-1}AP$. I have two binary ($0,1$) matrices $A$ and $B$. When does there exist a $P$ such that $A^t = P^{-1}AP$ and $...
- 363
10
votes
1
answer
3k
views
Reverse Minkowski (and related) Determinant Inequalities
For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:
$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$
and
$$\det(A+B+C) + \...
- 716
3
votes
0
answers
70
views
Condition number after some "non standard" transform
Given a positive definite matrix $A$, and a diagonal matrix $B$ with positive diagonal entries, is the following inequality generally true?
$$\kappa((A + B)(I + B)^{-1}) \leq \kappa(A)$$
$I$ is an ...
6
votes
2
answers
236
views
Bounding the non-multiplicativity of isometric projection
Every $A \in \text{GL}_n(\mathbb{R})$ has a unique Polar decomposition:
$A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$.
In particular the orthogonal factor is given by $$O_A=A(\...
- 6,741
5
votes
1
answer
2k
views
Upper bound of the largest eigenvalue of a PSD block matrix in terms of blocks
Let $\mathbf A=\left[\begin{matrix}\mathbf A_{11}&\mathbf A_{12}\\ \mathbf A_{21}&\mathbf A_{22}\end{matrix}\right]$ be a positive semi-definite matrix, $\mathbf A_{ij}\in\mathbb C^{n\times n}...
- 103
1
vote
0
answers
496
views
Reverse-mode Hessian matrix of the Cholesky factor
Let $ f(L) $ be a scalar function of the lower triangular Cholesky decomposition of the covariance matrix $ \Sigma $ such as $ \Sigma = LL' $.
Let's assume that we know the first and second derivative ...
- 11
4
votes
0
answers
749
views
When the integral of the product of two matrix exponentials is singular?
Let $A$ and $B$ be two $n \times n$ real matrices. (In my application, $A$ and $B$ are $6\times 6$ traceless singular real matrices) I am interested in finding the smallest $T$ such that the integral $...
- 41
1
vote
1
answer
462
views
Is this a full rank matrix? [closed]
According to the answer of znt to the previous version, I revise the question as follows:
Is there a real $(n-1)\times n$ matrix $A$
such that $A$ is not a full rank matrix and satisfy $a_{ii}&...
- 356
1
vote
0
answers
947
views
max min of ratio of quadratic forms
Consider the optimization over two vectors $x$ and $y$
$$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$
for two positive definite matrices $A$ and $B$.
This problem can be ...
1
vote
3
answers
195
views
The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra
Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X)
\otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ ...
- 356
1
vote
0
answers
56
views
Matrix diagonalization after a completely positive transformation
I have a hermitian matrix $A$ which can be diagonalized:
$$A=UDU^+,$$
where U is the unitary matrix and D is the diagonal matrix.
Next, I have a completely positive transformation over it, which is ...
- 171
1
vote
1
answer
607
views
The state-transition-matrix of a physical system,
Here's a simple but potential research problem that I am learning about.
Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
- 1
0
votes
1
answer
246
views
$P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?
Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and
$P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable.
$Z$ is eigenvalue of $P(Z )$ if $\...
- 151
26
votes
1
answer
1k
views
Real square roots of symmetric matrices
In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
- 5,186
2
votes
0
answers
125
views
When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?
We have $p \in \mathbb{N}$ and $\mathcal{H}$ is a Hilbert space.
let's consider a matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p} \in \mathcal{S_H}^{p\times p}\!\!\,,$ that is a $p\times p$...
- 195
6
votes
0
answers
587
views
Lower bound on the sum of singular values for a sum of Hermitian matrices
Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
- 907
1
vote
0
answers
137
views
Boundary of pseudospectra
Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
- 151
4
votes
0
answers
676
views
Weyl-type inequality for non-Hermitian matrices?
What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
- 6,541
1
vote
1
answer
940
views
Uniqueness and invariance of the LDLT decomposition
A real symmetric positive semi-definite matrix $A$ can be decomposed in the form
$A = P^TLDL^TP$,
where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
- 101
7
votes
2
answers
4k
views
Induced matrix norm less than one for matrices with spectral radius less than one
Let $A$ be a square matrix with elements in $\mathbb{R}$ or $\mathbb{C}$,
$\rho\left(A\right)$ stands for the spectral radius of $A$, i.e.,
the maximum absolute eigenvalue of $A$; $A^{*}$ is the ...
1
vote
0
answers
96
views
Let $M = \frac{1}{2}(A + {A^T})$ be real symmetric nonnegative matrix. Why does $\rho (A) \le {\lambda _{\max }}(M)$?
Let $A \in M_n$ be nonnegative, and consider the real symmetric nonnegative matrix
$M = \frac{1}{2}(A + {A^T})$.
Why does $\rho (A) \le {\lambda _{\max }}(M)$?
- 11
1
vote
0
answers
114
views
A question on Perron–Frobenius theorem [closed]
Let $A \in M_n$ is nonnegative(all $a_{ij}\ge0$).
Suppose $A$ has a nonnegative eigenvector(all entries$\ge0$ ) with $r ≥ 1$ positive entries and $n − r$ zero entries.
Why is there a permutation ...
- 11
9
votes
3
answers
4k
views
Fast Upper Triangular Matrix Exponentiation
Let $Q_n$ be a $n\times n$ matrix with $Q_n=\begin{pmatrix} -\lambda_1-\mu_1 & \lambda_1 & 0 & \cdots\\ 0 & -\lambda_2-\mu_2 & \lambda_2 & \cdots\\ \vdots & \vdots & \...
- 4,952
7
votes
1
answer
250
views
approximate stationary distributions of a doubly stochastic matrix and its supports
Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...
- 363
12
votes
0
answers
218
views
Which ordering of factors is needed to obtain this kind of determinantal inequalities?
Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
- 13.4k
21
votes
1
answer
2k
views
Almost commuting unitary matrices
Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...
- 901
-1
votes
1
answer
173
views
finding a unitary submatrix inside a random matrix
Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be ...
- 482
1
vote
0
answers
72
views
Triangle inequality for nonconvex functions of singular value vector
For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave ...
- 31
2
votes
0
answers
476
views
What does similar eigenvectors and eigenvalues of two matrices really mean? [closed]
Empirically I've noticed that diagonally dominant matrix G and it's diagonal version D (diagonal elements of G on the diagonal and all other elements are set to zero) produce similar eigenvalues and ...
- 29
4
votes
2
answers
202
views
Integral roots of circulant matrix
When does the circulant matrix have only integral roots?
For example: all roots of the adjacency matrix of the complete graph $K_n$ are integer, which its adjacency matrix is circulant, but in case ...
8
votes
2
answers
426
views
Existence and characterization of transitive matrices?
We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following:
For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw &...
- 720
0
votes
0
answers
84
views
Can we increase spectral norms of All maximum size square submatrices by orthogonal perturbation?
Let the matrix $A$ consist of $k$ columns from some $n \times n$ orthogonal (unitary) matrix. It is obvious that there is no perturbation of $A$ which
leaves its columns orthonormal,
increases ...
- 1