All Questions
187 questions
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 $n \times n$ ...
- 61
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
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 $...
- 917
6
votes
0
answers
489
views
Symmetric matrices with $\rho(A)\gg\|A\|_\infty$
For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
- 23k
5
votes
1
answer
319
views
Analytical form for the nuclear norm of an $n \times n$ matrix
I get the follow equation in a paper. Let $A \in \mathbb{R}^{2 \times 2}$, then $M = A^TA$ is a positive semi-definite matrix, the nuclear norm of $A$ is:
$$ \Vert A \Vert_* = \sqrt{\operatorname{tr}(...
- 71
5
votes
2
answers
495
views
Hadamard product and inertia
One of Schur's famous results says that if $A,B$ are positive semidefinite matrices, then the Hadamard (i.e. entrywise) product $A \circ B$ is also positive semidefinite. It's also true if "semi" is ...
- 6,962
5
votes
1
answer
405
views
Best orthogonal approximation of rank 1 matrix
Let $X=\lambda_0u_0v_0^T\in\mathbb{R}^{n\times n}$ be a rank 1 matrix where $\lambda_0\in\mathbb{R}$, $u_0,v_0$ are of unit Euclidean norm. What is the solution of the following problem?
$$\hat{X}=\...
- 1,495
5
votes
1
answer
2k
views
Inverse of a matrix and the inverse of its diagonals
While researching a question, I faced with the following problem: I have to prove that for a positive definite matrix we have
$${\mathbf n}^T {\mathbf R}^{-1}{\mathbf n}\geq {\mathbf n}^T {\mathbf D}...
- 273
5
votes
3
answers
693
views
Norm of triangular truncation operator on rank deficient matrices
Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ A\|\le\frac{\...
- 153
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 ...
- 51
5
votes
1
answer
910
views
Proving a majorization inequality for the singular value of the product of two matrices without using tensor product
For any two matrices $\mathbf{A},\mathbf{B} \in \mathbb{C}^{n \times n}$, we know that the following majorization inequality holds
$$
\tag{1}
\label{grz}
\sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \...
- 115
5
votes
2
answers
429
views
Simultaneous maximization of two Generalized Rayleigh Ritz Ratios
Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
- 1,421
5
votes
1
answer
2k
views
Diagonalization of real symmetric matrices with symplectic matrices
Consider the following real symmetric matrix
$M=\left[\begin{array}{ccc} A & B\\ B^T & D \end{array}\right]$
Both A and D are real symmetric $n\times n$ matrices. B is a real $n\times n$...
- 51
4
votes
3
answers
3k
views
Is this inequality involving the Frobenius norm right?
Let $A$ be a generic (or varying) square, real $ n \times n$ matrix. Let $G$ be a fixed $n \times k$ matrix, $k < n.$ Denote by $||.||_F$ the Frobenius norm.
Is it true that $||AG||_F \geq c(G) ||...
- 1,467
4
votes
2
answers
311
views
Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$
For given $n\times n$ real symmetric positive definite matrices $X_{1}, X_{2}$, let $Y := X_{1}^{-1}X_{2}$, and let $I$ be the identity matrix.
I would like to solve the following equation for the ...
- 1,538
4
votes
3
answers
239
views
Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$
Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix
$$
X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}.
$$
Such ...
- 2,712
4
votes
3
answers
4k
views
upper bounds on a certain matrix norm
Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?
- 6,962
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
votes
1
answer
206
views
How to find the analytical representation of eigenvalues of the matrix $G$?
I have the following matrix arising when I tried to discretize the Green function， now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute ...
- 153
4
votes
1
answer
721
views
Singular value decomposition of truncated discrete Fourier transform matrix
Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that
\begin{align}
F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N.
\end{align}
What we can say about the singular value ...
- 287
4
votes
1
answer
413
views
Lipschitz property of matrix function only depending on singular values
Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{...
- 1,495
4
votes
1
answer
289
views
A property of positive matrices
Let $\mathbb{S}^{dn} \ni X \succeq 0$ with $d,n \in \mathbb{N}$, where $X \succeq 0$ indicates that $X$ is positive semidefinite. Now partition $X$ into the block form
\begin{gather}
\begin{pmatrix}
...
- 81
4
votes
0
answers
990
views
Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
4
votes
0
answers
1k
views
Can an orthogonal matrix move monotonically toward a signed permutation matrix?
The question is motivated by this question on Mathematics SE.
Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
- 209
4
votes
0
answers
284
views
Maximizing a certain eigenvalue ratio
Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
- 2,712
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. $$
Let $...
- 195
4
votes
0
answers
676
views
Weyl-type inequality for non-Hermitian matrices?
What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
- 6,541
3
votes
1
answer
1k
views
Explicit formula for Cholesky factorization in a special case
I have a positive definite matrix of the form $Q+sI-\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the all-ones matrix), where $Q$ is "nice", nonnegative and known. I'd like to know if there is a ...
- 6,962
3
votes
1
answer
381
views
Hadamard $\ell_p$ sum of two symmetric positive semidefinite matrices
Does there exist $p>1$ such that for all $n\geq 2$, if $(a_{ij})$ and $(b_{ij})$ are symmetric positive semidefinite $n\times n$ matrices and $a_{ij}, b_{ij}\geq 0$ then $\bigl(\|(a_{ij},b_{ij})\|...
- 175
3
votes
1
answer
172
views
Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$
Every matrix $A\in M_4(\mathbb{R})$
can be represented in the form of $$A=\sum_{i=1}^{n(A)} B_i\otimes C_i$$ for $B_i,C_i\in M_2(\mathbb{R})$.
What is the least uniform upper bound $M$ for such $n(A)$...
- 356
3
votes
1
answer
296
views
Upper bounds on elements of a matrix
During my research I have come across matrices this type
$$C=B\left(B^T B\right)^{-1}B^T\ ,$$
where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...
- 265
3
votes
1
answer
2k
views
Relation between Frobenius norm, infinity norm and sum of maxima
Let $A$ be a sequence of $n \times n$ matrices so that the Frobenius norm squared satisfies $\|A\|_F^2 \simeq n$ and the infinity norm squared is $\|A\|_{\infty}^2 = 1$. Is the following true?
$$\...
- 105
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 ...
- 225
3
votes
1
answer
741
views
Operator norm of difference of matrix decompositions
This question is in part related to a question that I have already posed.
Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
- 55
3
votes
1
answer
196
views
Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ the conjuagation action fixes the first column?
Let $A \in GL_n(\mathbb R)$ be fixed. Let us consider the conjugation action by $G \in GL_n(\mathbb R)$, i.e., $G^{-1}AG$. I would like to see a way to identify the matrices such that the action fixes ...
- 195
3
votes
1
answer
264
views
When is this matrix singular?
Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.
1) For $t_k=k$, what is the condition on $\...
- 2,239
3
votes
1
answer
144
views
On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
- 61
3
votes
0
answers
2k
views
Multiplication of two Pauli string
Given a Pauli string $P_i \in \{ I,X,Y,Z\}^{\otimes n} $
Example: $P_0 = XXYIZ = X \otimes X \otimes Y \otimes I \otimes Z $.
Here $I,X,Y,Z$ are Pauli matrices defined explicitly as:
$$
I = \begin{...
- 131
3
votes
0
answers
244
views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
- 2,712
3
votes
0
answers
630
views
Diagonal elements of Hermitian matrices with given eigenvalues
Given real vectors $d = (d_1, \ldots, d_n)$ and $\lambda = (\lambda_1, \ldots, \lambda_n)$, where I will assume that their coefficients are arranged in non-increasing order, the Schur-Horn theorem ...
- 31
3
votes
0
answers
56
views
Equivalence Classes of a Subgroup of Similarity Transformations
Let $X$ be a real, finite-dimensional vector space and $A, B, C,$ and $D$ be matrices on $X$. I'm interested in the similarity classes of the block matrices
$$
\begin{bmatrix}
A & B\\
C & D\\
...
- 263
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
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 ...
3
votes
0
answers
611
views
Can anyone help me deduce a matrix inequality?
The following lemma is taken from references firstly.
Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$
for all $F$ satisfying $F^{...
- 103
3
votes
0
answers
193
views
Method to Generate Random Mutually Orthogonal Unitary Matrices
The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
- 133
3
votes
0
answers
125
views
Copositivity in matrix pencils
Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...
- 6,962
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
2
votes
1
answer
498
views
Does the Perron vector maximize $x^TAx$ in the simplex?
Let $\mathbf{A}$ be any $n\times n$ symmetric positive matrix ($A_{ij}>0$). It is easy to show that the solution to the following optimization problem
\begin{align}
\max_{\mathbf{x}}~~\mathbf{x^...
- 1,421
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 ...
- 2,246
2
votes
1
answer
74
views
Limitation through the singular values
Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...
- 143