# 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.

522
questions

**2**

votes

**0**answers

43 views

### 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 ...

**2**

votes

**0**answers

105 views

### 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(...

**0**

votes

**0**answers

34 views

### Schaten p norm of block matrices

Let $A=D\oplus 0$ be a diagonal Hermitian matrix and $B$ is an invertible Hermitian matrix with $(1,1)$ block being $B_{11}$ and $B_{11}$ and $D$ have the same dimensions. Then is it true that if $(1+|...

**6**

votes

**1**answer

209 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 ...

**6**

votes

**2**answers

145 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 ...

**0**

votes

**0**answers

23 views

### Is the SVD optimal (by Eckart Young theorem) because maximal data variance is captured from both the column and row space?

I am asking this question seeking validation of an intuitive understanding of the veracity of the Eckart-Young theorem which struck me in my study of the SVD and Principle Component Analysis. The ...

**2**

votes

**0**answers

74 views

### 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 $n\times n$ ...

**1**

vote

**1**answer

42 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 ...

**1**

vote

**1**answer

64 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 ...

**0**

votes

**0**answers

15 views

### Difficult matrix inequality stemming from possible lipschitz equivalence of two metrics

I am interested whether there exists a constant $C\geq 1$ such that
$\frac{\langle (x x^*)^{1/2}, (y y^*)^{1/2} \rangle}{\frac{1}{2}(||x||_2^2+||y||_2^2)}\geq C \frac{||x^*y||_1}{\frac{1}{2}(||x||_2^...

**0**

votes

**0**answers

41 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 \...

**0**

votes

**1**answer

118 views

### 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 ...

**2**

votes

**1**answer

52 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 \...

**3**

votes

**1**answer

272 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 +...

**2**

votes

**1**answer

69 views

### On conditionally negative definiteness of a kernel

We are interested in the kernel $\phi\left( x-y\right) =\frac{1}{2}\log\left( \cosh\left( x-y\right)
\right) -\log\left( \cosh\left( \frac{x-y}{2}\right) \right)$ on $\mathbb{R} \times \mathbb{...

**1**

vote

**0**answers

34 views

### 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 ...

**3**

votes

**0**answers

55 views

### 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),\...

**0**

votes

**0**answers

44 views

### 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 ...

**1**

vote

**1**answer

119 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). $AA^{\#}=A^{\#}A$ ...

**1**

vote

**1**answer

82 views

### 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 $\lambda_1\geq\lambda_2\geq \dots \geq \lambda_n$ ...

**2**

votes

**0**answers

66 views

### 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:
$$D^2u\geq - \dfrac{(1+K^2)^{1/2}}{\epsilon}I$$
...

**1**

vote

**0**answers

84 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 ...

**1**

vote

**0**answers

38 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 ...

**0**

votes

**0**answers

57 views

### Matrix Lie algebra: Semisimple element = semisimple matrix?

I am considering the set
$$ \mathbb{L} = \lbrace X \in \mathbb{C}^{2n \times 2n} \; ; \; J^{-1}X^HJ = -X \rbrace, \quad J = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix},$$ of complex ...

**2**

votes

**0**answers

61 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_{...

**2**

votes

**1**answer

74 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 ...

**5**

votes

**2**answers

243 views

### 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}) \...

**0**

votes

**0**answers

81 views

### Numerical range of tensor product of two matrix

Let $T\in M_n$. Is the following true
$$\bigcap\limits_{B\in M_2\\\text{tr}(B)=0}\left\{X\in M_2: W(X)\subseteq W(T)\text{ and } W(B\otimes X)\subseteq W(B\otimes T)\right\}\subseteq\bigcap\limits_{...

**1**

vote

**0**answers

34 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 ...

**1**

vote

**0**answers

54 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}=...

**2**

votes

**1**answer

97 views

### 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 ...

**8**

votes

**3**answers

536 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 $\lambda_1,\dots,\lambda_n$...

**1**

vote

**0**answers

75 views

### 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 $i$. ...

**20**

votes

**1**answer

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}\...

**6**

votes

**0**answers

533 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 $f$ ...

**10**

votes

**0**answers

638 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 ...

**0**

votes

**0**answers

37 views

### Sum of a two dimensional arithmetico-geometric suite

I am trying to compute the first column $P_{k,1}^n$ of the power of matrix $P^n$ where $P$ is a lower bidiagonal matrix with terms :
$$P_{i,j} = \left\{\begin{array}{cc}
i\alpha & \text{if }i=j,\\
...

**2**

votes

**0**answers

32 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
$$\...

**0**

votes

**1**answer

37 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
$$\mathbf{A}=\bar{\mathbf{A}}\mathbf{\Gamma},$$
...

**0**

votes

**1**answer

67 views

### 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\...

**2**

votes

**0**answers

96 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 ...

**0**

votes

**0**answers

60 views

### 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, ...

**8**

votes

**0**answers

180 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 ...

**0**

votes

**0**answers

107 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 ...

**1**

vote

**1**answer

87 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 ...

**6**

votes

**1**answer

407 views

### 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 ...

**0**

votes

**0**answers

94 views

### Numerical range of a 2-by-2 matrix

Let $X\in M_2(\mathbb{C})$ s.t. $W(X)\subseteq\text{Conv}\{\lambda_1,\cdots,\lambda_k\}$ and $\Vert X\Vert\leq\text{max}_{1\leq j\leq k}\vert\lambda_j\vert$. Then does it true that there exist $H_1, ...

**2**

votes

**0**answers

101 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 ...

**4**

votes

**2**answers

158 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}^{...

**0**

votes

**0**answers

82 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\...