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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20
votes
1answer
1k 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}\...
16
votes
2answers
2k 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 $...
8
votes
5answers
648 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 ...
6
votes
1answer
3k views

Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
7
votes
4answers
3k views

The derivative of the Cholesky Factor

Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be it's Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am ...
15
votes
3answers
881 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(...
6
votes
1answer
963 views

Comparing norms on tensor products of matrices

Given a Hilbert space $H$, let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T \Vert_1 = \sum_{j\geq 1} |s_j| $$ where $(s_1,...
4
votes
0answers
442 views

Generalizing Autonne-Takagi factorization

Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that: A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
4
votes
2answers
1k views

How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$ Note that the perturbative calculation of square root ...
3
votes
1answer
257 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})\|...
2
votes
0answers
78 views

Hadamard $\ell_p$ sum of two symmetric positive semidefinite matrices: follow-up

I asked the following question here: "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}\...
1
vote
1answer
398 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}&...
35
votes
3answers
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 ...
25
votes
2answers
40k 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 ...
27
votes
1answer
2k 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 $G$,...
31
votes
5answers
1k 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 ...
12
votes
0answers
198 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^{...
10
votes
1answer
2k 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) + \...
16
votes
1answer
1k 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 ($C=\bf 0$) $...
13
votes
7answers
890 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 ...
15
votes
1answer
723 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{...
13
votes
2answers
2k 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 $H$? ...
11
votes
1answer
739 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 $\| \cdot \|_1$ ...
10
votes
0answers
626 views

Two questions around the $abc$-conjecture

Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers. The abc-conjecture can be formulated using these two metrics as: For ...
4
votes
0answers
380 views

A minimal eigenvalue inequality

Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
4
votes
3answers
2k views

Comparison of the L_p norm of a matrix and its entry-wise absolute value

Suppose $A_{n \times n}$ is a matrix and $A' = (|A_{ij}|)$ is its entry wise absolute form, can be give an upper bound and lower bound of the L_p norm $\|A\|_p$ using the L_p norm of the absolute ...
2
votes
1answer
162 views

A question on the partial sum of infinite doubly stochastic matrix

Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Is the following statement true ? $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^na_{ij} >0 $$ Any reference or comment on this is ...
13
votes
2answers
817 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 ...
9
votes
2answers
2k 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 \...
4
votes
1answer
350 views

On a vanishing integral inner product

Let $G(z)$ be an $n\times m$ rational matrix-valued function of full column rank on the unit circle. Further, let $P(z)$ be an $m\times m$ rational matrix-valued function positive definite on the unit ...
4
votes
4answers
2k views

The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq \...
7
votes
3answers
485 views

Trace of a nonlinear matrix equation (cont'd)

Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration $$ X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star) ...
6
votes
2answers
551 views

Matrix-convexity of inverse of the cofactor matrix

Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ ...
5
votes
1answer
333 views

Trace of a nonlinear matrix equation

Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration $$ X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star) ...
4
votes
2answers
285 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 ...
3
votes
2answers
1k views

Maximizing trace of $\mathrm V^T \mathrm A \mathrm V$ for $\mathrm A$ symmetric (alternate proof with min/max-theorem)

I'm trying to work out a proof for the following proposition: Let $A \in \mathbb{R}^{n,n}$ a real, symmetric matrix with eigenvalues $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n$, then $$\...
3
votes
1answer
372 views

The Matrix form of the Van der Pol equation

Motivated by the classical Van der Pol equation which has a unique periodic attractor, we consider the following differential equation on $M_{2}(\mathbb{R})\times M_{2}(\mathbb{R}):$ $$(*)\;\;\;\...
8
votes
1answer
770 views

Is the p-norm of a matrix strictly log-convex?

Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is, $$ \|A\|_p = \sup_{x\neq 0}\frac{\|Ax\|_p}{\|x\|_p}...
5
votes
1answer
344 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 ...
5
votes
1answer
175 views

Is a pointwise “ simple tensor” valued continuous map a tensor product of two continuous maps?

A matrix $A\in M_{4}(\mathbb{C})$ is called a simple tensor if $A=B\otimes C$ for two $2\times 2$ matrices $B,C$. Assume that $X$ is a Hausdorff topological space.Assume that $f:X\to M_{4}(\...
5
votes
3answers
462 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{\...
4
votes
2answers
155 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}^{...
4
votes
1answer
350 views

Equivalent metrics on symmetric positive definite matrices

By similar arguments as for the proof of the golden-thompson inequality (see "Log majorization and complementary Golden-Thompson type inequalities" by T.Ando and F.Hiai) we can show that for all A,B ...
3
votes
2answers
677 views

How to solve a non-homogeneous quadratic matrix equation?

I am looking to solve the following matrix equation for $G$ $$GHG + M = 0$$ where $G$, $H$, and $M$ are square, symmetric, real matrices. $H$ is negative-definite and $M$ is positive-definite. $G$ ...
2
votes
2answers
257 views

The structure of the $n$-th power of a special matrix

Assume the following matrix $$ C_p^{(a,b)}:=\left( \begin{array}{cccccc} a &a &0 &\cdots &\cdots &0 \\ 0 &0 &a &\ddots &\ddots &\vdots \\ \vdots &\ddots &...
1
vote
0answers
276 views

What is the Birkhoff norm of a Perron vector?

Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector? By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$. P.S. This is ...
0
votes
1answer
541 views

Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows \begin{eqnarray*} ||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} \...