The study of real and complex matrices and their algebraic and analytical properties, including: eigenvalues and eigenvectors, positive definite matrices, matrix inequalities, invariant subspaces, perturbation analysis, matrix functions.

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1answer
179 views

Nonnegative Matrix

Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ...
2
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0answers
231 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
3
votes
3answers
374 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 ...
1
vote
1answer
150 views

Weighted Spectral l-2 norms arising from matrix inner products

The spectral $l^2$ norm of a complex matrix is given by: $\|A\|= \left( \sum_{k=0}^{N-1} s_k(A)^2 \right)^{1/2}$ where $s_k(A)$ are the singular values of $A$ ordered so as to be non decreasing in ...
2
votes
1answer
140 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 ...
0
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1answer
91 views

Norm bound on eigen-vector change caused by rank-one update

Suppose $A$ is a positive semi-definite, Hermitian matrix with a unit-norm eigen vector $\textbf{v}$ corresponding to its largest eigen value $\lambda$. Let $B = A + \alpha \textbf{z}\textbf{z}^H$, ...
1
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0answers
88 views

All Unitarily Invariant Minkowski Norms on $\mathfrak{su}(N)$

It is well known that the some matrix norms satisfying: $\|A\| = \|UAV\|$, $\forall A \in \mathbb{C}^{N \times N}, \forall U,V \in U(N)$ are the Schatten-p norms: $\|A\|_p= \left( \sum_{k=0}^{N-1} ...
3
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1answer
95 views

Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background. Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables $x^{(1)},\ldots,x^{(n)}$. We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
7
votes
1answer
776 views

A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity: Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...
2
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0answers
106 views

A - B is semidefinite, what the relationship about their eigenvalues? [closed]

$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...
5
votes
3answers
370 views

Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$: $$ \mathrm{arg}\max_R ...
2
votes
1answer
95 views

Dimension independent computational complexity of singular value decomposition

Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$). Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time which is ...
0
votes
1answer
188 views

Bounding the positive semi-definite matrix with its block diagonal matrix [closed]

Can we bound $\mathbf{A}$ with $\mathbf{A^*}$ as ${\bf{A}} \preceq {{\bf{A}}^*}$ where \begin{equation} {\bf{A}} = \left[ {\begin{array}{*{20}{c}} {{{\bf{A}}_{11}}}&{...}&{{{\bf{A}}_{1N}}}\\ ...
3
votes
2answers
251 views

On a determinant inequality of positive definite matrices

Assume that $B$ and $A$ are two positive definite matrices. Take $B^*$ a block diagonal matrix with block $B_{11}$ and $B_{22}$ of $B$. This means the following: $$ B=\left[\begin{array}{ll} ...
2
votes
1answer
200 views

More than controlability: Speed of controllability!

Consider the continuous linear time-invariant system $$ \begin{array}{l} \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\ \mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t) ...
1
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0answers
75 views

M-matrix with nonconstant entries properties

I have a matrix $J(x)$ with $J_{ij}(x)=f_{ij}(x)$ where vector $x$ is $x=x_1, x_2, ..., x_m$. I have shown that $J(x)$ is an M-matrix for all $x$. There is known review paper by Plemmons (1977) of 40 ...
1
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1answer
466 views

Reachability in graphs using adjacent matrix

Assuming a graph $G$ with $N$ nodes distributed in a $\mathcal{L}\times\mathcal{L}$ area randomly. There is an edge between two nodes if and only if the Euler distance between them is equal or less ...
1
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0answers
104 views

Row subset selection of matrix to optimize condition number

Given a matrix $\mathbf{A} \in \mathbb{C}^{N\times M}$ with $N \gg M$. This matrix results from a linear equation system and has a certain structure (however, I do not think that details provide any ...
6
votes
1answer
266 views

Best rank one approximation

Assume $u,v\in\mathbb{C}^n$ are complex vectors. I was wondering if there is a closed form expression for the following problem in terms of $u$ and $v$ \begin{equation*} \arg\min_{x\in\mathbb{C}^n} ...
-3
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1answer
166 views

adjacency matrix of random geometric graphs [closed]

Consider a graph with N nodes. All nodes are distributed as a Poisson point process with density of λ in a L*L area. There is an edge between two nodes if and only if the distance between them is less ...
6
votes
1answer
342 views

The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as $H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$ What is known about the singular values $\sigma_1\geq\ldots\geq \sigma_n$ of $H$? For example, ...
4
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1answer
365 views

An inequality involving operator and trace norms

Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...
1
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0answers
85 views

Copositivity in matrix pencils

Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...
0
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2answers
149 views

Matrix-Norm aquivalence with p-Norm [closed]

Let $A$ be a square Matrix and $||\cdot ||_p$ the induced Matrixnorm for $1 \leq p \leq \infty$. Is it true that $$||A||_p\leq \max(||A||_1,||A||_{\infty})?$$ For $p=2$ the answer is yes because ...
8
votes
1answer
221 views

Inequalities for Hadamard products of complex symmetric matrices

Consider a complex symmetric matrix $$ C= C_R + i C_I $$ with $C_R,C_I \in \text{Mat}_{n\times n}(\mathbb R)$ symmetric, and assume that the eigenvalues of $C_R$ are all strictly positive. Then, $C$ ...
1
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0answers
121 views

Estimate the diagonal of a Cholesky factor…?

I'm computing several hundred Cholesky factorizations of large, sparse matrices, and I'm really only doing Cholesky factorization because I need to know the diagonal elements of the Cholesky factor L. ...
0
votes
2answers
166 views

positive semidefiniteness: a psd matrix substracted by another rank 1 psd matrix

Given that $A$ is a positive semidefinite matrix, $x$ is a vector, $\lambda_0 \in [0, +\infty) $ is a real non-negative number. I want to know the answer to the following optimization problem. $$ ...
0
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0answers
301 views

Diagonal of the inverse of a 6x6 symmetric partitioned matrix

Let $$M = \begin{bmatrix} A & B \\ B & C \end{bmatrix}$$ in which $A$, $B$ and $C$ are $3 \times 3$ matrices being also symmetric. In fact, they are quite similar, just differing on a single ...
4
votes
2answers
159 views

Maximal norm-1 projection

Suppose I have a real unitary matrix $U$ and a unit vector $\mathbf{x}, \|\mathbf{x}\|_2 = 1$. What is the solution to the following problem? $$ \widehat{\mathbf{x}} = \arg\max_{\mathbf{x}, ...
0
votes
2answers
232 views

Eigenvalues of an amplification matrix

Let $A$ and $B$ square real matrices. I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1. Can we say something about the eigenvalues ...
0
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0answers
64 views

Big eigenvalues of a special stochastic matrix

Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq ...
1
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0answers
59 views

Which matrix/operator in a cone has the largest negative spectral part?

Background: Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where ...
9
votes
1answer
411 views

M-matrix plus S-matrix is P-matrix?

I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two matrices and I would ...
11
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2answers
384 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 ...
1
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0answers
80 views

Matrices with a common fischer basis

Let $A$ be a real symmetric $n\times n$ matrix normalised such that $Tr[A]=1$. Define 'fischer basis' as the basis in which all diagonal elements are equal to $\frac{1}{n}$. The motivation for this ...
0
votes
1answer
623 views

How to compute difference between 2 similarity matrices?

Hello, I have two n*n correlation matrices with values ranging between -1,1. (2 correlation matrix because I have the same n terms under 2 different conditions) I then transformed the correlation into ...
3
votes
2answers
255 views

A short question about the DFT matrix

Is the DFT matrix the unique* unitary matrix with all entries of same magnitude? (*up to some trivial transformations)
4
votes
1answer
225 views

Rank of a matrix with missing entries

Let $M$ be a $2^n \times 2^n$ matrix over real number field, where the rows and columns are indexed by subsets of $[n] := \{1,2,\ldots,n\}$, and defined as follows, $ M_{A, B} = 1 $ if $A \subseteq ...
4
votes
2answers
249 views

system of homogeneous matrix equations

Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$. One of my friend asked me the following ...
16
votes
1answer
637 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 ...
9
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2answers
858 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, ...
2
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1answer
221 views

Asymptotic Behavior of Non-Analytic Function of the Eigenvalues

Hello, Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$. If $A_n$ were a sequence of Hermitian ...
3
votes
1answer
254 views

Singular values of the sum of A and A^T

Dear all, As a part of my research, I need to achieve a lower bound to the smallest singular value, $\sigma_{n}\(A+A^{T}\)$ for a stochastic $A$ (as a function of the singular values of $A$), which ...
1
vote
0answers
123 views

Kernel of modified Kronecker sum

The Kronecker sum of two matrices $A \in M(n \times n, \mathbb{R})$ and $B \in M(m \times m,\mathbb{R})$ is defined by the matrix $$A \oplus B = A \otimes I_m + I_n \otimes B \in M(nm \times nm, ...
3
votes
2answers
114 views

a monotone relation for s-numbers

Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one ...
4
votes
2answers
326 views

Optimization version of the Sylvester equation

The Sylvester equation is a matrix equation of the form $AX-XB=C,$ where $A,B,C$ are given matrices of dimension $m\times m,n\times n$ and $m\times n$ and $X$ is an unknown matrix of dimension ...
2
votes
1answer
158 views

Number of matrices of a given rank satisfying this condition

Let $A_1$ and $A_2$ be two arbitrary $n\times n$ matrices with entries in $Z_p$. How many $n\times n$ matrices $B$ are there so that both $A_1-B$ and $A_2-B$ are of rank $n-1$ or less? What is the ...
3
votes
2answers
144 views

Triangularizing a function matrix with smooth eigenvlaues

Given a matrix with function entries, which are smooth and homogeneous, and having smooth eigenvalues, can we find a conjugating matrix with smooth and homogeneous entries that triangularize the given ...
5
votes
1answer
308 views

A matrix inequality involving the Hilbert-Schmidt norm

This question comes from a problem in PDEs on which I'm currently working. Let $a$ be a $3\times 3$ matrix, real symmetric and positive definite. Denote with $\|a\|^2 _ 2=\sum a_{ij}^2$ the square of ...
0
votes
1answer
202 views

Bounding a determinant ratio

Let $A=[A_{0}\ E;E^{T} \ B]$ be a real positive definite matrix and let $B$ be a principal submatrix. I am interested in tightly bounding $\frac{|B|}{|A|}$ from below in some "explicit" way that will ...