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21 votes
1 answer
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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 $A_j'$ ...
Omid Hatami's user avatar
13 votes
1 answer
1k views

A generalization of the Powers-Stormer inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...
Henry Yuen's user avatar
  • 2,019
12 votes
2 answers
1k views

Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?

Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite. My question is: Is there a ...
Omar's user avatar
  • 123
8 votes
3 answers
691 views

Commutant of the conjugations by unitary matrices

Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
Jochen Glueck's user avatar
7 votes
1 answer
511 views

Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\trace{trace}$Let $A \in \SL(2,\mathbb{R})$ and $\trace(A)>2$. Is it true that $$\lVert A\rVert \leq \lVert A^2\rVert,$$ where $\lVert \rVert$ is the ...
Adam's user avatar
  • 1,043
7 votes
0 answers
355 views

An $L^{\infty}$ version of principal component analysis?

I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal. I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
floc's user avatar
  • 193
6 votes
1 answer
277 views

Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$

Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...
Fedor Petrov's user avatar
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) ||...
Learning math's user avatar
3 votes
2 answers
939 views

Positive definiteness of infinite tridiagonal matrices

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \...
Giuseppe Negro's user avatar
3 votes
0 answers
115 views

Recovering the matrix when the Schur decomposition of its blocks are known

Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and $$E=\left(\begin{array}{cc} G & X \\ X^t & H \end{array}\right)$$ where $G,H,X$ are $m\times m$ matrices. Suppose that $...
ABB's user avatar
  • 4,058
3 votes
0 answers
275 views

Schur-Horn theorem for principal submatrices

The Schur-Horn theorem says that there exists a Hermitian matrix with diagonal entries $d_1,d_2,\ldots,d_n$ and eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n$ if and only if $(\lambda_1,\lambda_2,\...
Nathaniel Johnston's user avatar
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!
Vincent Russo's user avatar
2 votes
1 answer
215 views

Forming real positive semidefinite matrices from complex matrices

I have asked this question on the Mathematics Stack Exchange: https://math.stackexchange.com/questions/4924554/forming-real-symmetric-positive-semidefinite-matrices-from-complex-matrices. Let $Q \in \...
Mthpd's user avatar
  • 31
2 votes
1 answer
112 views

The eigenvectors of adding a particular rank one matrix to the circulant matrix

Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$. Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
ABB's user avatar
  • 4,058
2 votes
1 answer
346 views

Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators. We want to look at matrices that agree in most of their entries and ...
user avatar
2 votes
0 answers
43 views

Selecting some linearly independent columns of a particular matrix

Let us consider the matrix $C=A_1+A_2$ where : $A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$ $A_2$ is the the $n$ by $n$ block ...
ABB's user avatar
  • 4,058
2 votes
0 answers
246 views

Decay rate of least eigenvalue of Gram matrices

Consider the Hilbert space $H=L^2_w(I)$ as the weighted $L^2$ space, where $I\subseteq\mathbb{R}$: $$ L_w^2(I)=\{\phi:I\rightarrow\mathbb{R}:\,\|\phi\|^2=\int_I \phi(x)^2w(x)\,dx<\infty\}. $$ In ...
user39756's user avatar
  • 141
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} < ...
Learning math's user avatar
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 $p\times p$...
Obriareos's user avatar
  • 195
0 votes
1 answer
75 views

The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$

Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix. $$...
ABB's user avatar
  • 4,058
0 votes
0 answers
55 views

Johnson-Lindenstrauss type result for matrix factorization

The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
optimal_transport_fan's user avatar
0 votes
0 answers
124 views

Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same

Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with $$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$ How can we compute the eigenvectors of $T$?
ABB's user avatar
  • 4,058
0 votes
0 answers
174 views

Lipschitz map on positive definite cone of $n$-by-$n$ matrices

A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
Reza's user avatar
  • 91
0 votes
0 answers
198 views

eigenvalues of the product of a unitary with a diagonal

In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
ABB's user avatar
  • 4,058
0 votes
0 answers
90 views

Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?

Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that $$\tag{1}S^{-1}\cdot A\cdot S = \...
fsp-b's user avatar
  • 463