Questions tagged [matrix-analysis]

The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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Is this proof of Perron's theorem correct, and if so is it original?

A few years ago, I came up with this proof of Perron's theorem for a class presentation: https://pi.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf I've written an outline of it below ...
Hannah Cairns's user avatar
35 votes
3 answers
3k 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 ...
M. Lin's user avatar
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35 votes
6 answers
2k views

Trigonometry / Euclidean Geometry for natural numbers?

Let $d(a,b) = 1 - \frac{2\gcd(a,b)^3}{ab(a+b)}$ be a metric on natural numbers without $0$. The metric space $X = \{x_0,x_1,\cdots,x_n\},n>2$ is isometric embeddable in $\mathbb{R}^n$ if and only ...
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34 votes
2 answers
66k views

Proof for the derivative of the determinant of a matrix [closed]

I was looking for theorems that might be helpful in order for some proofs that I have and I came across the following one: $$\frac{d}{dt} [\det A(t)]=\det A(t) \cdot \operatorname*{tr}[A^{-1}(t)\cdot ...
Max's user avatar
  • 489
29 votes
1 answer
3k views

Is there an explicit formula for the hessian of "Determinant"?

Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function. Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$,...
Ali Taghavi's user avatar
29 votes
2 answers
5k views

Consequences of eigenvector-eigenvalue formula found by studying neutrinos

This article describes the discovery by three physicists, Stephen Parke of Fermi National Accelerator Laboratory, Xining Zhang of the University of Chicago, and Peter Denton of Brookhaven National ...
Ivan Meir's user avatar
  • 4,782
26 votes
1 answer
1k views

Finding the closest matrix to $\text{SO}_n$ with a given determinant

$\newcommand{\GLp}{\operatorname{GL}_n^+}$ $\newcommand{\SLs}{\operatorname{SL}^s}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\Sig}{\Sigma}$ $\newcommand{\id}{\text{Id}}$ $\newcommand{\...
Asaf Shachar's user avatar
  • 6,611
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 ...
Lev Borisov's user avatar
  • 5,166
25 votes
1 answer
2k views

The abc-conjecture as an inequality for inner-products?

The abc-conjecture is: For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have: $$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...
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24 votes
0 answers
1k views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
Mark M. Wilde's user avatar
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 $A_j'$ ...
Omid Hatami's user avatar
20 votes
2 answers
4k views

Does the matrix exponential preserve the positive-semi-definite ordering?

Suppose $A$, $B$, are symmetric, real valued matrices and $B-A$ is positive-semidefinite, i.e. $A≼B$. Does that imply $e^A ≼ e^B$? Would love some intuition here. I know for instance that $A≼cI \iff ...
nikka's user avatar
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20 votes
1 answer
25k views

When does the $4\times 4$ 'false Sarrus rule' compute the determinant correctly?

This question is most probably not research level, but I thought that the MO folks might like it... Feel free to close. Here is the motivation: If you have ever taught a maths course for engineers ...
Dirk's user avatar
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19 votes
4 answers
814 views

How much redundancy resides in an $n \times n$ orthogonal matrix?

Suppose one has an $n \times n$ orthogonal matrix $M$: $$ \left( \begin{array}{ccc} 0.239326 & 0.846726 & 0.475161 \\ 0.768893 & 0.13356 & -0.625272 \\ 0.592897 & -0....
Joseph O'Rourke's user avatar
17 votes
1 answer
3k views

2x2 subdeterminants of a matrix

If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B. Given then all these 2x2 determinants of an ...
Carlo Mantegazza's user avatar
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 ($C=\bf 0$) $...
Wolfgang's user avatar
  • 13.2k
16 votes
2 answers
3k views

Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns $...
Igor Rivin's user avatar
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16 votes
2 answers
3k views

The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as follows $$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$$ What is known about the singular values $\sigma_1 \geq \cdots \geq \sigma_n$ of $H$? ...
alext87's user avatar
  • 3,167
16 votes
0 answers
373 views

An inequality for matrix norms

Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically: Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
Mostafa's user avatar
  • 4,454
15 votes
7 answers
2k views

Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?

(Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.) To make my problem more understandable, I start with the ...
Adam's user avatar
  • 345
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 & \...
amcalde's user avatar
  • 301
15 votes
3 answers
1k views

Is this lower bound for a norm of some complex matrices true?

Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$. I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(...
Mahdi's user avatar
  • 1,991
15 votes
1 answer
858 views

Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix. For complex numbers $a_1,\ldots,a_n$, I will use the notation $$ \mbox{...
Eusebio Gardella's user avatar
14 votes
5 answers
4k views

Matrix trace & norm [closed]

For any nonnegative semidefinite matrix $A$ and any matrix $B$, we have $$\mbox{tr} (AB) \le \mbox{tr} (A) \, \|B\|$$ where $\mbox{tr}(\cdot)$ is the trace and $\|\cdot\|$ is the operator norm. How ...
Ivanov's user avatar
  • 157
14 votes
4 answers
2k views

Vandermonde matrix is totally positive

A totally positive matrix $M\in \mathcal{M}_{n\times m}(\mathbb R)$ is such that all of its minors of all sizes are positive. It is true that any Vandermonde matrix (with well-ordered positive entries)...
Loïc Teyssier's user avatar
14 votes
2 answers
570 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 ...
Wolfgang's user avatar
  • 13.2k
13 votes
5 answers
1k views

Packing obtuse vectors in $\mathbb{R}^d$

I came across this attractive theorem: Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that form an obtuse angle with one another. This was proved1 as a corollary of a lemma about ...
Joseph O'Rourke's user avatar
13 votes
2 answers
912 views

Computing a large permanent

Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix? I have tried to use the matlab function written by Luke Winslow which works great for smaller matrices ...
Felix Goldberg's user avatar
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
12 votes
2 answers
3k views

On the positive definiteness of a linear combination of matrices

In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated. QUESTION: Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
Tatin's user avatar
  • 895
12 votes
4 answers
823 views

Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positive real parts

Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix ...
PAb's user avatar
  • 187
12 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
  • 1,899
12 votes
0 answers
749 views

Does this matrix norm inequality have interesting application in other areas of mathematics?

In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices: Theorem 3. ‎Let $A=[a_{ij}]$ be a real symmetric ...
Mostafa's user avatar
  • 4,454
12 votes
0 answers
457 views

More mysterious properties of Gram matrix

This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question. The following fact could be extracted from 0402087: For any $a_i\...
Daniil Rudenko's user avatar
12 votes
0 answers
217 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^{...
Wolfgang's user avatar
  • 13.2k
12 votes
0 answers
312 views

Ratio of entries of A and log A where A is a triangular matrix

Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - 1)!}}$...
Johann Cigler's user avatar
11 votes
2 answers
772 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(...
Nikolayevich's user avatar
11 votes
4 answers
5k views

The derivative of the Cholesky factor

Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be its Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am ...
Steven Pav's user avatar
11 votes
2 answers
3k views

Do singular values dominate eigenvalues?

Suppose $A$ is an $n \times n$ complex matrix with singular values $s_1 \ge s_2 \ge \cdots \ge s_n$ and eigenvalues $(\lambda_i)_{i=1}^{n}$ arranged so that $|\lambda_1| \ge |\lambda_2| \ge \cdots \...
David Callan's user avatar
  • 1,125
11 votes
2 answers
2k views

Inverse of a small submatrix

Let $A$ be a large matrix (say, $1000 \times 1000$), and let $\mathcal I = \{2,3,5\}$ be a set of row/column indices. Let $(A^{-1})_{\cal I \times I}$ denote the submatrix of $A^{-1}$ that consists of ...
John Smith's user avatar
11 votes
2 answers
606 views

Multiplicative Identity for all elements in SU(n)

Let $\{P_i\}$ be a subset of $SU(n)$ such that for any $U$ in another subset (or perhaps subgroup) $H$ of $SU(n)$: $$P_1UP_2U\cdots P_mU=I$$ where $I$ is the identity element. Is there a sequence $\{...
Kaveh Khodjasteh's user avatar
11 votes
2 answers
571 views

Nearby matrices have nearby leading eigenvectors?

Suppose I have a symmetric positive semidefinite matrix $A$ with leading eigenvalue $1$ of multiplicity $1$ and remaining eigenvalues $\leq\epsilon$. I am told that another symmetric positive ...
Dustin G. Mixon's user avatar
11 votes
1 answer
954 views

Show that these vectors are linearly independent almost surely

So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question. Problem: I have $m<n$ real $...
FeedbackLooper's user avatar
11 votes
1 answer
550 views

Is there a "formula" for the point in $\text{SO}(n)$ which is closest to a given matrix?

$\newcommand{\Sig}{\Sigma}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\distSO}[1]{\dist(#1,\SO)}$ $\newcommand{\distO}[1]{\text{dist}(#1,\On)}$ $\newcommand{\tildistSO}[1]{\operatorname{...
Asaf Shachar's user avatar
  • 6,611
11 votes
2 answers
752 views

A (linear) optimization problem subject to (linear) matrix inequality constraints

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
Ludwig's user avatar
  • 2,682
11 votes
0 answers
303 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,...
Ozzy's user avatar
  • 383
10 votes
2 answers
1k views

When the sum of positive definite matrices converges, does the sum of the norm of the associate matrices converges?

Suppose $A_k>0$ (which means they are positive definitive square $n\times n$-matrices with $n>1$). If $\sum_{k=1}^\infty A_k$ exists, then $\sum_{k=1}^\infty \|A_k\| < +\infty$, Where $\|A\|=\...
Xifeng Su's user avatar
  • 173
10 votes
1 answer
577 views

is it possible to have two non-isomorphic non-regular graphs with the same adjacent spectrum and the same laplacian spectrum?

For two regular graphs $G$ and $H$, it is possible for them to share the same adjacent spectrum and the same laplacian spectrum. While, on the other hand, is it possible to have two non-regular graphs ...
josephS's user avatar
  • 103
10 votes
2 answers
717 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. ...
vansy's user avatar
  • 143
10 votes
1 answer
465 views

(Asymmetric) matrix power series in closed form: $\sum_{i=0}^{\infty} A^i \left(A^i\right)^{\top}={?}$

Let $A\in \mathbb{S}^{N\times N}$ be a symmetric, real and stable matrix, i.e., $\rho(A)<1$, where $\rho(A)$ stands for the spectral radius of $A$. Then, $$\sum\limits_{i=0}^{\infty} A^{2i}=\left( ...
Augusto Santos's user avatar

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