**17**

votes

**1**answer

1k 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**

votes

**2**answers

1k 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**

votes

**1**answer

237 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

**1**answer

306 views

### Singular values of the sum of A and A^T

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 is generally ...

**1**

vote

**0**answers

175 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

**2**answers

115 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

**2**answers

453 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

**1**answer

203 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

**2**answers

152 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

**1**answer

526 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

**1**answer

356 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 ...

**4**

votes

**5**answers

581 views

### Spectrum of transition matrix for symmetric random walk

I asked this question previously on math.stackexchange.com, where it had little traction.
Consider the symmetric random walk on $\{0,1,…,n\}$ with transition probabilities $P(j→j±1)=1/2$ for $0 ...

**1**

vote

**1**answer

165 views

### bounding the sum of the entries of the inverse of a 0-1 matrix away from 1?

Let $A\in\mathbb{R}^{n\times n}$ be an invertible 0-1 matrix. Is it possible that the sum $a:=\sum_{i,j}(A^{-1})_{ij}$ of entries of $A^{-1}$ is not equal to 1, but exponentially close (w.r.t. $n$) to ...

**1**

vote

**1**answer

655 views

### power of a block triangular matrix

I have a matrice in the form :
$$M =
\begin{pmatrix}
A & 0 & 0 \\\
B & A & 0 \\\
C & D & A
\end{pmatrix}
$$
where $A,B,C,D$ are diagonalizable square matrice and I want to ...

**5**

votes

**2**answers

309 views

### Simultaneous maximization of two Generalized Rayleigh Ritz Ratios

Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...

**2**

votes

**1**answer

660 views

### Trace inequality for matrices with determinant 1

Let $A$ and $B$ be two matrices with $\det(A)=\det(B)=1$. Does it follow that
$\sqrt{\mathrm{tr}(A^TB^TBA-I)}\le\sqrt{\mathrm{tr}(A^TA-I)}+\sqrt{\mathrm{tr}(B^TB-I)}$
I suspect that this can be ...

**0**

votes

**1**answer

460 views

### Simultaneous Jordanization

Hello everyone
I would like to have a detailed reference to the statement bellow:
Let $A,B\in \mathbb{R}^{n\times n}$ such that $AB=BA$. Suppose $A$ has real eigenvalues only and $B$ is ...

**4**

votes

**2**answers

1k views

### Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.

Consider a random matrix $\mathbf{A} \in \mathbb{C}^{N \times N}$ of rank $m$ with $m < N$ that follows the Wishart distribution ( http://en.wikipedia.org/wiki/Wishart_distribution ).
I have a ...

**1**

vote

**3**answers

1k 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 ...

**0**

votes

**2**answers

390 views

### a question about the Jordan form [closed]

Some reference say that if rank($A$)=rank($A^2$),then the geometric and algebraic multiplicities of the eigenvalues $\lambda=0$ are equal;that is,all the Jordan blocks correspondint to $\lambda=0$ (if ...

**1**

vote

**3**answers

687 views

### Number of parameters needed to specify a Hermitian matrix of rank r.

Hi,
i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature.
1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank ...

**0**

votes

**3**answers

556 views

### Convex Combination of 2 hermitian matrices

Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$ (both matrices ...

**5**

votes

**4**answers

4k views

### Eigenvalues of infinite matrices [closed]

I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can ...

**1**

vote

**2**answers

610 views

### can eigenvector be found without computing the eigenvalue [closed]

Is there any ways to compute the eigen vector without computing explicitly the associated eigenvalue?
Actually, I'd like to compute the largest eigenvalue of a positive matrix from its eigen vector, ...

**1**

vote

**1**answer

306 views

### Does this sequence converge to zero?

Description
Let $\{e_n\}$, $e_n\in \mathbb{R}^p$ be a sequence of vectors, $\{U_n\}$, $U_n\in\mathbb{C}^{p\times p}$ be a sequence of unitary matrices (that is $U_i^*=U_i^{-1}$, $^*$denonts conjugate ...

**2**

votes

**4**answers

186 views

### Nonlinear eigenvalue problem - sorta

Suppose you have an equation of the form $Ax=f(x)$, where $A$ is a $n \times n$ matrix, $x$ is a vector of length $n$ and $f(\cdot)$ is some function. Is there a name for this sort of problem?

**3**

votes

**1**answer

948 views

### norm of (sub)stochastic matrix

Is there any bounds for the norm of sub-stochastic matrix? (But it's not doubly stochastic matrix, I mean only the row sum is less than 1, while the column sum may not.

**3**

votes

**2**answers

262 views

### Hadamard product and inertia

One of Schur's famous results says that if $A,B$ are positive semidefinite matrices, then the Hadamard (i.e. entrywise) product $A \circ B$ is also positive semidefinite. It's also true if "semi" is ...

**1**

vote

**0**answers

90 views

### a weighted sum of Hermitian matrices and selection of weight values

We have $N$ Hermitian matrices $A_i$ and $N$ weight values $w_i$, $1\leq i\leq N$, $\sum_{i=1}^N w_i=1$.
Then we can obtain a new Hermite matrices $\sum_{i=1}^N w_iA_i$. let us assume $\lambda$ is ...

**16**

votes

**2**answers

1k 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 ...

**6**

votes

**2**answers

573 views

### Exponentiating 4 by 4 matrix analytically

Does there exist an analytical method by which i can exponentiate a 4 by 4 matrix, in the same way as the general 2 by 2 matrix case in pauli matrix basis. I have dirac matrices (which are composed of ...

**1**

vote

**0**answers

136 views

### An $L^{\infty} Version of Principal Component Analysis?

I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal.
I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...

**5**

votes

**0**answers

689 views

### When is the inverse diagonally dominant?

There is a large literature devoted to studying the inverses of diagonally dominant matrices. I'd like to know if there is information about a so-to-say opposite situation: we have a matrix $A$ and ...

**1**

vote

**0**answers

178 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 ...

**3**

votes

**2**answers

771 views

### Woodbury formula

I wonder - do you know of any example where the Woodbury formula (cf. http://en.wikipedia.org/wiki/Woodbury_matrix_identity) was crucially used to prove anything?
It might be a useful computational ...

**1**

vote

**0**answers

296 views

### Diagonalizing matrix with a special conjugate transpose property

Hi all,
I'm looking for the minimum criterion on $A\in M_{3x3}(\mathbb{C})$ (a $3x3$ complex matrix) such that:
1) $A$ is diagonalizable by a matrix $T\in M_{3x3}(\mathbb{C})$
2) $T$ is such that ...

**0**

votes

**0**answers

152 views

### vector equation

Suppose you have an equation of the form $Hx=Ky$, where $x,y$ are vectors of length $n,m$ respectively ($m>n$) and $H,K$ are matrices of orders $n \times n,n \times m$ respectively. Is there some ...

**2**

votes

**2**answers

411 views

### A sum of eigenvalues

Let $X$ be an $n\times n$ symmetric matrix. Suppose $\lambda_1(X)\geq \lambda_2(X) \geq \cdots \geq \lambda_n(X)$ are eigenvalues of $X$. Let $r$ be any integer with $1\leq r\leq n$. It is well-known ...

**2**

votes

**1**answer

199 views

### Simultaneous decomposition of three projectors

A projector $P$ is a Hermitian matrix satisfying $P^2=P$. For any two projectors, it is easy to show that there exists a unitary matrix $U$ such that both $U^*PU$ and $U^*QU$ are block-diagonal ...

**1**

vote

**1**answer

385 views

### sign-flipping inverse

Consider this matrix:
$Z=\begin{bmatrix}23.9 & -7 & -17 \\\\ -7 & 23.9 & -17 \\\\ -17 & -17 & 33.9 \end{bmatrix}$
Its inverse is entrywise negative (you can check...) and ...

**3**

votes

**3**answers

2k views

### upper bounds on a certain matrix norm

Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?

**6**

votes

**2**answers

365 views

### Triangularizing a matrix with function entries

Hi Everybody!
Given a matrix, with smooth functions as arguments is there any result which say about its triangularization?
I know that, the question is in affirmative for diagonalizing a matrix ...

**2**

votes

**1**answer

186 views

### Eigenvalue estimation by Lyapunov's method

I have seen somewhere the following results related to Lyapunov equation:
Let $A\in \mathbb{R}^n$ be a stable matrix in the sense of having negative real part eigenvalues. Let $\Re\lambda()$ denote ...

**1**

vote

**2**answers

384 views

### matrix stability criterion

I have a $5 \times 5$ parametric nonnegative matrix and want to show that it's stable (in the sense that all eigenvalues are positive). It is not symmetric, but I do know in advance that it has 5 real ...

**1**

vote

**2**answers

361 views

### Sufficient conditions for inverse-positivity

I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in Explicit formula for Cholesky factorization in a special case, but the ...

**2**

votes

**1**answer

931 views

### Explicit formula for Cholesky factorization in a special case

I have a positive definite matrix of the form $Q+sI-\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the all-ones matrix), where $Q$ is "nice", nonnegative and known. I'd like to know if there is a ...

**1**

vote

**0**answers

175 views

### matrix-theoretic terminology query

Is there an accepted term for the following property?
Let $A$ be a real matrix such that all entries of the eigenvector corresponding to the least eigenvalue have the same sign.
NOTES: (1) The case ...

**2**

votes

**1**answer

187 views

### Generalizing the spectral radius of a unistochastic matrix

Consider a square matrix $A$, and from it construct $B$ whose entries are the squared magnitudes of those in $A$. What can we say about the spectral radius of $B$? I know that for a unitary matrix ...

**2**

votes

**1**answer

376 views

### Null Space Perturbations

Hi,
I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases.
The distilled version of the ...

**1**

vote

**1**answer

228 views

### Bandwidth reduction of multiple matrices

Suppose I have a symmetric matrix A, and several diagonal matrices $D_1,D_2,...$
Are there any matrix transformations, such as $P^\top A P$ so that $P^\top AP$, $P^\top D_1 P$, $P^\top D_2 P$, etc ...