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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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}) \...
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Prove the optimal solution to maximizing nuclear norm with constraints is attained at corner points of feasible region

The nuclear norm (trace norm) of a matrix $X \in \Bbb R^{m \times n}$ is defined as $$\|X\|_* := \sum_{i=1}^{\min(m,n)} \sigma_i(X)$$ where $\sigma_i(X)$ are the singular values of $X$. The ...
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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 $...
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2 answers
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A question of invertibility of matrices

Let $A$ and $B$ be self-adjoint $n \times n$ matrices. Let $A$ be diagonal. Suppose $A+tB$ and $tA+B$ are invertible for all $t \in \mathbb R$. What can we say about $A$ and $B$? My guess is that $\...
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1 answer
400 views

Solving equation of matrix valued functions

Given $n\times n$ matrices with entire functions entries (holomorphic on all of the complex plane $\mathbb{C}$) $A(z)=[a_{ij}(z)],B(z)=[b_{ij}(z)]$, i.e., $a_{ij}(z),b_{ij}(z)$ are entire functions ...
MathGuest's user avatar
2 votes
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177 views

Bounding the condition number of a matrix associated with an even symmetric positive definite function

Define a set $A = \{x_i/x_i\in\mathbb{R}^m, i = 1,2,3..n\}$. Let $f:\mathbb{R}^m\to(0,\infty)$ be an even symmetric positive definite function. Let $D = [d_{i,j}]$ be an $n\times n$ matrix such that $...
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Does there always exist a matrix satisfying certain tracial conditions

Given odd integers $0<a<b$, I want to know if there exists an $n$ by $n$ real valued square matrix $M$ such that $$ M_{ij} = M_{ji} \quad \forall i,j \in \{1,2\dots n\}$$ $$ \sum_{i=1}^n M_{ij} =...
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Riemannian metrics on matrix space for which the restriction of trace function to each complete geodesic is a bounded function

Edit: According to comment by Leo Monsaingeon I revise my question: Is there a Riemannian metric on $M_n(\mathbb{R})$ for which the function $trace$ is a bounded function on every complete(whole)...
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3 votes
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Euclidean volume of symmetric matrices in operator norm

This is a nearly identical question to Euclidean volume of the unit ball of matrices under the matrix norm except in the symmetric case. Let $\mathrm{Sym}_{n \times n}(\mathbb{R})$ be the space of ...
steve's user avatar
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Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?

Consider the Frey-Hellegouarch curve given $a,b$ natural numbers: $$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$ Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(a+...
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6 votes
1 answer
416 views

Matrix inequality : trace of exponential of Hermitian matrix

I want to know whether the following inequality holds or not. \begin{align} (\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1} \end{align} where $A, B$ are Hermitian ...
s hukahi's user avatar
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2 answers
495 views

Reference request: continuity of Cholesky factor

It most books dealing with Cholesky decomposition, or it is variants, one finds a statement of the form if $A$ is symmetric $k\times k$ positive semi-definite (non-negative definite) then the $k\times ...
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An elementary proof of Davies' inequality

In the paper Lipschitz continuity of functions of operators in the Schatten classes, Davies proved the following matrix inequality. Let $a_i,b_i>0$ for $1\leq i\leq n$ and $A$ be an $n\times n$ ...
user07001129's user avatar
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1 answer
307 views

How can I find minimum and maximum eigenvalue of non-positive define matrix [closed]

There is a power iteration method, but it only returns the greatest(in absolute value) eigenvalue of matrix. So when we have negative eigenvalues it'll give wrong results. Is there any method, which ...
Dima Shkad's user avatar
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1 answer
969 views

$\arg\max$ in the dual norm of the nuclear norm

Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by $$\left \| X\right\| := \max\limits_{i \in \{1, \dots, \min\{m,n\}\} }\sigma_i (X)$$ whereas the nuclear norm is ...
Santiago Armstrong's user avatar
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46 views

"Probability" for a partitioned matrix to be singular

Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix $$ M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
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Matrix whose entries are given by polynomial $A_{ij} = p(\lambda_i, \lambda_j)$; when is it positive semidefinite?

Let $A$ be a matrix whose entries are given by a polynomial, $$ A_{ij} = p(\lambda_i, \lambda_j) $$ where $p(\lambda_i,\lambda_j) = p(\lambda_j,\lambda_i)$ is symmetric. Are there standard methods ...
Felix Huber's user avatar
2 votes
1 answer
330 views

Bound for matrix inner product based on singular values

Regarding the matrix inner product based on singular values, Lewis (1995) "The convex analysis of unitarily invariant matrix functions" states the result by von Neumann that $\langle X,Y \rangle \leq \...
Prateek Sasan's user avatar
3 votes
1 answer
402 views

Inequality for $AB + BA$ when $A,B\geq0$, reference request

Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues. It is well-known that the eigenvalues of the expression $AB +...
Felix Huber's user avatar
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Minimum rank of a product of two block diagonal matrices with an arbitrary matrix

Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
Juan's user avatar
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Regularity of Moore-Penrose pseudo-inverse

Let $k\in\mathbb{N}\cup\{0\}$, let $\Omega\subseteq\mathbb{R}^n$ be open, connected and let $G\in C^k(\Omega;\mathbb{R}^{n\times n})$ satisfy $$ \operatorname*{rank}G(x)= \operatorname*{rank}G(y),\...
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Upper bound on the condition number of the product of a random sparse matrix and a semi-orthogonal matrix

Let $G \in \mathbb{R}^{n \times m}$ (m > n, m = O(n)) whose all entries are i.i.d. distributed as $\mathcal{N}(0, 1) * \text{Ber}(p)$. Let $V \in \mathbb{R}^{m \times n}$ be a fixed semi-orthogonal ...
nikhil_vyas's user avatar
1 vote
1 answer
151 views

Commuting matrices of complex functions

If $A(z) :=[A_{ij}(z)] $ and $B(z) :=[B_{ij}(z)] $ are two invertible $n\times n$ matrices of entire complex valued functions entries $A_{ij}(z)$, and $B_{ij}(z) $ with (1). $AA^{\#}=A^{\#}A$ ...
Guest's user avatar
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1 answer
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Eigenvalues of adjacency matrix of a k-regular graph

If $A_G$ is the adjacency matrix of a k-regular graph, let $B = J+xA_G$, where J is the matrix whose elements are all 1s and $x\in R$ is a scalar. If $\lambda_1\geq\lambda_2\geq \dots \geq \lambda_n$ ...
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Estimates on norm Hessian Matrix

Let $u:\Omega \rightarrow \mathbb{R}$ a twice differential function, with $\Omega$ a subset of $\mathbb{R}^n$. Suppose that we have the following: $$D^2u\geq - \dfrac{(1+K^2)^{1/2}}{\epsilon}I$$ ...
Jason's user avatar
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0 answers
112 views

Full-rank Hadamard product given a certain structure

Let us assume that we have a full-rank randomly chosen $k\times (m\cdot l)$ matrix, $\boldsymbol{H}$, with $l \leq k \leq (m\cdot l)$ and no specific structure (e.g., a realization of an IID complex ...
Juan's user avatar
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1 vote
0 answers
123 views

Subgradient chain rule

Suppose $$F:\mathbb{R}^n \to \mathbb{R},\; F(x)=\mathrm{max}_\mathrm{eig}(C-\mbox{diag}(x)).$$ I am trying to find a subgradient of $F$ at $x_0$. A subgradient of $\mathrm{max}_\mathrm{eig}$ is given ...
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0 answers
266 views

Bound for the inverse of a summation of rank-1 matrices

Given vectors $x_1,\dots,x_T \in R^d$ satisfying $\|x_i\|_2 = 1$, define $A_0 = I$ and $A_t = I + \sum_{i=1}^tx_ix_i^\top$ for $t \geq 1$. We are interested in the following quantity: \begin{align} S_{...
Matrix9994's user avatar
2 votes
1 answer
86 views

Dubious matrix monotonicity

Coming from a problem in game theory, I arose at some dubious monotonicity like property for matrices of the following art. Let $H=\lbrace h\in\mathbb{R}^{n}\colon h_{1}+\dots+h_{n}=0\rbrace$. I'm ...
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2 answers
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Is this subset of matrices contractible inside the space of non-conformal matrices?

Set $\mathcal{F}:=\{ A \in \text{SL}_2(\mathbb{R}) \, | \, Ae_1 \in \operatorname{span}(e_1) \, \, \text{ and } \, \, A \, \text{ is not conformal} \,\}$, and $\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \...
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1 vote
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116 views

Spectral abscissa of symmetric matrix with skew-symmetric perturbation

I am interested in bounds on the minimal distance between the spectral abscissa $\max_{\lambda\in\sigma(A)}\mathrm{Re}\lambda$ of a matrix $A$ and the eigenvalues of its perturbated version $A+S$. In ...
user98563's user avatar
2 votes
0 answers
254 views

Eigenvector of Hadamard matrix functions

Let $X\in\mathbb{R}^{n\times n}$ with SVD $X=UDV^T$. Are there known results regarding the eigenvectors of $Y=X^{\odot g}$? I am mainly interested in simple functions such as $g(z)=z^2$, i.e. $Y_{ij}=...
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3 votes
1 answer
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Flatness directions of the operator norm

It is known that the standard operator norm $\|\cdot\|_2$ over ${\bf M}_n({\mathbb R})$ is very flat, as is any operator norm (= subordinated norm) actually. The set of extremal points of the unit ...
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8 votes
3 answers
636 views

Representation theorem for matrices (reference request)

Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that $$ A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k, $$ where $\lambda_1,\dots,\lambda_n$...
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1 vote
0 answers
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Vectors that satisfy $\sum_{i=1}^n Y_i X_i^\top = I$ and $\sum_{i=1}^n \frac{1}{p_i}Y_iY_i^\top = \Sigma(P)^{-1}$

Let $X_1,\dots,X_n$ be vectors in $\mathbb{R^d}$. Assume all of the vectors are inside the unite $\ell_2$ ball. Let $P$ be a vector in the probability simplex $\Delta_n$ with $P_i>0$ for all $i$. ...
good bandit's user avatar
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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6 votes
0 answers
714 views

Spectral norm bound on smooth primary matrix function perturbation

Consider an $L$-Lipschitz function $f: \mathbb{R} \to \mathbb{R}$ (so $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$) and Hermitian PSD matrices $A, B \in \mathbb{C}^{n\times n}$. Define $f(A)$ to be $f$ ...
Ewin's user avatar
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10 votes
0 answers
781 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 ...
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2 votes
0 answers
70 views

Case of equality in entrywise spectral radius bound

Let $A,B$ denote square matrices such that $\lvert A_{ij}\rvert\le B_{ij}$ for all $i,j$, and denote the spectral radius by $\rho$. From the Gelfand spectral radius formula it is easy to see that $$\...
Julian's user avatar
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1 vote
1 answer
59 views

Effect of column normalization on maximum diagonal entry

Let $\mathbf{A}$ be a $M\times N$ complex matrix, and $\bar{\mathbf{A}}$ be constituted by normalizing each column of $\mathbf{A}$. Therefore, we have $$\mathbf{A}=\bar{\mathbf{A}}\mathbf{\Gamma},$$ ...
Math_Y's user avatar
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1 answer
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Choosing the best submatrix

Let $\mathbf{A}_{m\times n}$ be a matrix with non-negative elements. Assume that a submatrix $\mathbf{B}$ from $\mathbf{A}$ is defined as \begin{align} B_{i,j} = \begin{cases} A_{i,j}, & i\in\...
Math_Y's user avatar
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0 answers
125 views

Is the matrix $\mu_f(X_i \cap X_j)$ positive definite?

Let $X_1,\ldots, X_n$ be finite subsets of some larger finite set $Z$. Let $f:Z \rightarrow \mathbb{R}_{>0}$ be any function, and define a (counting) measure $\mu_f(X) = \sum_{x \in X} f(x)$ for a ...
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Eigenvalues of the matrix obtained by letting some of the rows vanish, hoping for some inequality

Let $A$ be an $n \times n$ matrix. Let $A_k$ be the matrix obtained by keeping the first $k$ rows of $A$ fixed and substituting $0$ for the rows $k+1$ to $n$. To be precise, we write $A= [R_1...R_k, ...
Learning math's user avatar
10 votes
1 answer
600 views

Minimum distance of a symmetric matrix to diagonal matrices

Let $A=(a_{ij})$ be an arbitrary $n \times n$ real symmetric matrix and $n\geq 2$. Let $\| \cdot \|$ denote the operator $2$-norm or equivalently the maximum absolute value of eigenvalues for ...
Mostafa's user avatar
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0 answers
364 views

Comparison of two similarity matrices

English is not my first language, so please excuse any mistakes. I'm working with two similarity matrices on the same data set: Suppose I have $n$ items, and I calculated the similarity of each item ...
Catasaur's user avatar
1 vote
1 answer
106 views

A unitary matrix of functions [closed]

If $A(z)=[A_{ij} (z)] $ is an $n\times n$ unitary matrix valued functions. Is there a characterization of such matrix if: (1) the entries are analytic functions on a set $D$. and (2) if the ...
Guest's user avatar
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7 votes
1 answer
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Operator norm of square root of matrix vs original

If I have a nonsymmetric matrix whose operator norm is $\leq 1$ and square root it, does its operator norm remain below $1$? More formally, I want to know whether there is always at least one square ...
user149538's user avatar
3 votes
0 answers
119 views

Distance between two algebraic sets

We are in $M_n(\mathbb{R})$ equipped with the Frobenius norm $||A||^2=tr(AA^T)$. Let $Z=\{(A,B)\in M_n(\mathbb{R})^2;A^2-AB-B^2=0\}$ and $T=O(n)^2$. It is easy to see that $Z\cap T=\emptyset$ and ...
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7 votes
3 answers
552 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
0 votes
0 answers
90 views

Special kind of translation and rotational invariance of the numerical range

Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Is the following statement true? If $W(B\otimes X)\subseteq W(B\otimes T)$ for any $B\in M_n$ then $W(B\otimes (X+I_n))\subseteq W(B\...
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