All Questions
187 questions
6
votes
2
answers
1k
views
Nontrivial lower bound on the sum of matrix norms
Let $X, V\in\mathbb{R}^{n\times r}$ such that $X^\top V$ is symmetric. The central quantity I care about is
\begin{equation}
\|XV^\top\|_{F}^2+\|X^\top V\|_{F}^2 +[\text{Tr}(X^\top V)]^2.
\end{...
- 515
-1
votes
1
answer
195
views
Determinant of $Z^TZ$ [closed]
If one is looking at the characteristic polynomial of the $m \times m$ dimensional matrix $Z^TZ$ then apparently the coefficient of $(-1)^{m-k}$ in it can be written as, $\sum_{U \subset [m], V \...
- 20.2k
11
votes
2
answers
777
views
Trace of non-commutable matrices
Let $M_1$ and $M_2$ be two symmetric $d\times d$ matrices. What is the relationship between
$tr(M_1M_2M_1M_2)$ and $tr(M_1^2 M_2^2 )$?
P.S. I tried a few examples and found
$$
tr(M_1M_2M_1M_2) \le tr(...
- 96.4k
4
votes
0
answers
245
views
On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix
(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...
- 195
11
votes
0
answers
313
views
Jaffard's theorem - finite matrices
For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies
$$
A(k,l) \leq C (1+\left|k-l\right|)^{-r},
$$
for some $C>0$,
then
$$
A^{-1}(k,...
- 2,545
6
votes
1
answer
777
views
Is every real matrix conjugate to a semi antisymmetric matrix?
Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with ...
- 356
1
vote
2
answers
486
views
Closed form for integral of function of a symmetric positive definite matrix
Let $M$ be a real symmetric positive definite matrix of size $n \times n$, and let $\log M$ denote its (principal) matrix logarithm.
Is it possible to evaluate the following integral in closed form?
...
- 21.5k
9
votes
2
answers
894
views
Inverse of special upper triangular matrix
Consider the following $n \times n$ upper triangular matrix with a particularly nice structure:
\begin{equation}\mathbf{P} = \begin{pmatrix}
1 & \beta & \alpha+\beta & \dots & (n-3)\...
7
votes
1
answer
433
views
Smallest singular value of $X\mapsto AX^{T}+XA^{T}$
Question: Given the long and skinny matrix $A\in\mathbb{R}^{m\times n}$ with $m\ge n$, define the matrix valued operator
$$\mathcal{A}:X\mapsto AX^{T}+XA^{T}.$$
What is the tightest nontrivial lower-...
- 148k
6
votes
1
answer
1k
views
Inequality between nuclear norm and operator norm for positive definite matrices
I will use $\|\|_*$ to denote the nuclear norm (sum of singular values) and $\|\|_2$ to denote the operator norm / matrix 2-norm (largest singular value).
Consider two positive definite $n \times n$ ...
- 24.2k
2
votes
2
answers
123
views
Behavior of matrix rank under thresholding of its elements
Let $A$ be a $n \times n$ real matrix of, say, rank $r$. Consider the matrix $$\max \{0,A\}$$ whereby each negative element of $A$ is set to $0$ and the non-negative elements are left unchanged. Is ...
14
votes
2
answers
574
views
A simple but curious determinantal inequality
Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices and $k>0$ real. Then $A^k$ is well-defined and experimentally, we have $$\det(A^k+BABA^{-1})\geqslant \det(A^k+BA^{-1}BA),$$or ...
- 28.6k
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 ...
- 28.6k
3
votes
0
answers
56
views
Equivalence Classes of a Subgroup of Similarity Transformations
Let $X$ be a real, finite-dimensional vector space and $A, B, C,$ and $D$ be matrices on $X$. I'm interested in the similarity classes of the block matrices
$$
\begin{bmatrix}
A & B\\
C & D\\
...
- 263
7
votes
1
answer
2k
views
Bound the eigenvalue of product of matrices?
Let $H$ be a $n \times n$ real symmetric matrix that has eigenvalues with absolute value less than 1. Define the matrix $M = \prod_{i=1}^n (I - e_ie_i^{\top}H)$ where $e_i$ denotes the $i^\text{th}$ ...
CommunityBot
- 1
8
votes
1
answer
490
views
Determinants (and traces) of linear maps of matrices
Let $k$ be a field or a commutative ring with unit and let $F:M_n(k)\to M_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A_1XB_1 + \cdots + A_m X B_m$ for some $A_i,B_i\in ...
- 883
1
vote
1
answer
437
views
Eigenvector of a nonnegative matrix in closed form
Consider $n\times 1$ vector $\alpha = (\alpha_{1}, ..., \alpha_{n})$, where $0<\alpha_{i}<1$, and $\sum_{i=1}^{n}\alpha_i = 1$. Construct the $n\times n$ zero-diagonal matrix $A$ with $(i,j)$-th ...
- 1,538
1
vote
1
answer
623
views
Can I modify the singular values of a matrix in order to get a negative eigenvalue?
Let $A \in \mathbb{R}^{n \times n}$ be a real nonsymmetric matrix with eigenvalues $\left\{\lambda_i : i=1..n\right\}$ with positive real part $\Re(\lambda_i) > 0$ $\forall i=1..n$
Let $A=U\Sigma ...
- 54.2k
8
votes
2
answers
362
views
Unitary factor in polar decompositions
Let $A, B$ be $n$-square (Hermitian) positive definite matrices. Let $AB=U|AB|$ be the polar decomposition of $AB$. So $U$ is unitary (called the unitary factor of $AB$). What is the optimal constant $...
- 356
6
votes
0
answers
587
views
Lower bound on the sum of singular values for a sum of Hermitian matrices
Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
- 159
15
votes
3
answers
3k
views
Determinant of a $k \times k$ block matrix
Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \...
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_{...
- 54.2k
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
35
votes
3
answers
4k
views
A curious determinantal inequality
In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...
- 1,748
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
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 & \...
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 ...
CommunityBot
- 1
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 ...
- 54.2k
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 ...
- 1
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
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
$$\det(A+B+C) + \...
CommunityBot
- 1
3
votes
0
answers
70
views
Condition number after some "non standard" transform
Given a positive definite matrix $A$, and a diagonal matrix $B$ with positive diagonal entries, is the following inequality generally true?
$$\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(\...
CommunityBot
- 1
1
vote
1
answer
608
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 ...
- 1,167
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
3
votes
1
answer
451
views
Singular values of the sum of A and A^T
As a part of my research, I need to achieve a lower bound to the smallest singular value, $\sigma_{n}(A+A^{T})$ for a stochastic $A$ (as a function of the singular values of $A$), which is generally ...
- 52.3k
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 $M_{n}(\mathbb{C})$ ...
- 2,665
10
votes
2
answers
733
views
Product $PVPVP$ is elementwise nonnegative?
Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative.
...
CommunityBot
- 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 $\...
- 570
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 ...
- 20.2k
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 ...
- 2,118
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?
- 1,590
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
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 $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
CommunityBot
- 1
-1
votes
1
answer
173
views
finding a unitary submatrix inside a random matrix
Let $\mathbf{R} \in \mathbb{C}^{~m \times n} $ with $m \leq n $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Let where $r$ be ...
- 482
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
CommunityBot
- 1
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 ...
- 4,784
1
vote
0
answers
148
views
Perturbation of eigenvalues of some special matrices
In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...
- 20.2k
5
votes
2
answers
480
views
Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices
I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
- 121