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.
715
questions
5
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1
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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
$$
\tag{1}
\label{grz}
\sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \...
0
votes
1
answer
655
views
Prove the optimal solution to maximizing nuclear norm with constraints is attained at corner points of feasible region
The nuclear norm (trace norm) of a matrix $X \in \Bbb R^{m \times n}$ is defined as
$$\|X\|_* := \sum_{i=1}^{\min(m,n)} \sigma_i(X)$$
where $\sigma_i(X)$ are the singular values of $X$.
The ...
4
votes
0
answers
1k
views
Can an orthogonal matrix move monotonically toward a signed permutation matrix?
The question is motivated by this question on Mathematics SE.
Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
3
votes
2
answers
370
views
A question of invertibility of matrices
Let $A$ and $B$ be self-adjoint $n \times n$ matrices. Let $A$ be diagonal. Suppose $A+tB$ and $tA+B$ are invertible for all $t \in \mathbb R$. What can we say about $A$ and $B$?
My guess is that $\...
4
votes
1
answer
400
views
Solving equation of matrix valued functions
Given $n\times n$ matrices with entire functions entries (holomorphic on all of the complex plane $\mathbb{C}$)
$A(z)=[a_{ij}(z)],B(z)=[b_{ij}(z)]$,
i.e.,
$a_{ij}(z),b_{ij}(z)$ are entire functions ...
2
votes
0
answers
177
views
Bounding the condition number of a matrix associated with an even symmetric positive definite function
Define a set $A = \{x_i/x_i\in\mathbb{R}^m, i = 1,2,3..n\}$. Let $f:\mathbb{R}^m\to(0,\infty)$ be an even symmetric positive definite function.
Let $D = [d_{i,j}]$ be an $n\times n$ matrix such that $...
2
votes
1
answer
403
views
Does there always exist a matrix satisfying certain tracial conditions
Given odd integers $0<a<b$, I want to know if there exists an $n$ by $n$ real valued square matrix $M$ such that
$$ M_{ij} = M_{ji} \quad \forall i,j \in \{1,2\dots n\}$$
$$ \sum_{i=1}^n M_{ij} =...
0
votes
1
answer
187
views
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)...
3
votes
0
answers
148
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 ...
7
votes
1
answer
438
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(a+...
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 ...
4
votes
2
answers
495
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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 ...
3
votes
0
answers
131
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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 $n\times n$ ...
1
vote
1
answer
307
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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 ...
2
votes
1
answer
969
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$\arg\max$ 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
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
$$
M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
0
votes
1
answer
142
views
Matrix whose entries are given by polynomial $A_{ij} = p(\lambda_i, \lambda_j)$; when is it positive semidefinite?
Let $A$ be a matrix whose entries are given by a polynomial,
$$
A_{ij} = p(\lambda_i, \lambda_j)
$$
where $p(\lambda_i,\lambda_j) = p(\lambda_j,\lambda_i)$ is symmetric.
Are there standard methods ...
2
votes
1
answer
330
views
Bound for matrix inner product based on singular values
Regarding the matrix inner product based on singular values, Lewis (1995) "The convex analysis of unitarily invariant matrix functions" states the result by von Neumann that $\langle X,Y \rangle \leq \...
3
votes
1
answer
402
views
Inequality for $AB + BA$ when $A,B\geq0$, reference request
Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues.
It is well-known that the eigenvalues of the expression $AB +...
1
vote
0
answers
58
views
Minimum rank of a product of two block diagonal matrices with an arbitrary matrix
Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
3
votes
0
answers
71
views
Regularity of Moore-Penrose pseudo-inverse
Let $k\in\mathbb{N}\cup\{0\}$, let $\Omega\subseteq\mathbb{R}^n$ be open, connected and let $G\in C^k(\Omega;\mathbb{R}^{n\times n})$ satisfy
$$
\operatorname*{rank}G(x)= \operatorname*{rank}G(y),\...
0
votes
0
answers
126
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
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). $AA^{\#}=A^{\#}A$ ...
1
vote
1
answer
1k
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
1k
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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:
$$D^2u\geq - \dfrac{(1+K^2)^{1/2}}{\epsilon}I$$
...
1
vote
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 ...
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 ...
2
votes
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_{...
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 ...
5
votes
2
answers
304
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}) \...
1
vote
0
answers
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 ...
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}=...
3
votes
1
answer
114
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
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 $\lambda_1,\dots,\lambda_n$...
1
vote
0
answers
94
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 $i$. ...
25
votes
1
answer
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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:
$$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...
6
votes
0
answers
714
views
Spectral norm bound on smooth primary matrix function perturbation
Consider an $L$-Lipschitz function $f: \mathbb{R} \to \mathbb{R}$ (so $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$) and Hermitian PSD matrices $A, B \in \mathbb{C}^{n\times n}$. Define $f(A)$ to be $f$ ...
10
votes
0
answers
781
views
Two questions around the $abc$-conjecture
Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers.
The abc-conjecture can be formulated using these two metrics as:
For ...
2
votes
0
answers
70
views
Case of equality in entrywise spectral radius bound
Let $A,B$ denote square matrices such that $\lvert A_{ij}\rvert\le B_{ij}$ for all $i,j$, and denote the spectral radius by $\rho$. From the Gelfand spectral radius formula it is easy to see that
$$\...
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
$$\mathbf{A}=\bar{\mathbf{A}}\mathbf{\Gamma},$$
...
0
votes
1
answer
86
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\...
3
votes
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 ...
0
votes
0
answers
95
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, ...
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 ...
0
votes
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 ...
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 ...
7
votes
1
answer
1k
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 ...
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 ...
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}^{...
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\...