**3**

votes

**1**answer

574 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

238 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

81 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 ...

**12**

votes

**2**answers

556 views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg n.$ 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

542 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

128 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

469 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

155 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

634 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

256 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

151 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

391 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

178 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

317 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

909 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

349 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

177 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

322 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

324 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

691 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

174 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

149 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

330 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

208 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 ...

**4**

votes

**2**answers

354 views

### Perron Frobenius with one negative pair of entries

Suppose you have a real symmetric matrix $A$ which is positive except for $a_{ij},a_{ji}$, who are negative.
While it is not generally true that the eigenvector of the dominant eigenvalue of $A$ is ...

**1**

vote

**4**answers

633 views

### A matrix diagonalization problem

For matrices $X,Y\in [0,1]^{n\times m}$, for n > m, is there a square matrix $W\in R^{n\times n}$ so that $X^TWY$ is diagonal if and only if $Y = X$? Furthermore, $X$ and $Y$ are column normalized so ...

**2**

votes

**1**answer

1k views

### How about eigenvalues of a positive matrix and a positive rank one matrix

Assume that A, B are positive n by n matrices and the rank of B is 1, B=xx*.
If the eigenvalues of A are a_1≥a_2≥...≥a_n, and x is not the eigenvector of A, then there are d_i≥0 such that eigenvalue ...

**1**

vote

**2**answers

4k views

### How to compare two similarity matrices?

Hi,
Suppose that I have two nxn similarity matrices. These matrices contain similarity information between n items. Although both matrices contain similarities of the same n items they do not contain ...

**1**

vote

**0**answers

4k views

### Eigenvalues of the sum of two matrices

Hello,
I know that given two matrices A and B, estimating the eigenvalues of A + B = C as a function of the eigenvalues of A and of the eigenvalues of B is generally a non-easy problem. I was ...

**4**

votes

**4**answers

1k views

### The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq ...

**5**

votes

**1**answer

451 views

### Rank of the absolute-value matrix $|M|$ vs. rank of $M$

Let $M$ be a real matrix of rank $r$ (and let us set $M=UV^T$, with $U,V^T\in\mathbb{R}^{n\times r}$, to fix the notation).
Let $|M|$ be the matrix obtained by taking the absolute value of each entry ...

**1**

vote

**0**answers

215 views

**0**

votes

**0**answers

295 views

### Optimization of a matrix with an objective function (for ML)

Hi.
I need to do max. likelihood for an objective likelihood function L (minimize it), and the target is a matrix. ie:
$$min_KL(K)$$
For example:
K is, let's say, of size 3x3 and with initial ...

**12**

votes

**2**answers

1k views

### Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?

Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...

**3**

votes

**2**answers

870 views

### Spectral properties of the LDL^T matrix factorization

Assume that a square, symmetric matrix $A$ can be factored into $A=LDL^T$ where $L$ is unit lower triangular and $D$ is diagonal. For indefinite $A$, $D$ may have $2x2$ blocks on the diagonal. How ...