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
292 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 ...
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3answers
669 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?
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2answers
327 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 ...
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1answer
161 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 ...
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2answers
276 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 ...
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2answers
306 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 ...
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1answer
533 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 ...
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0answers
173 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
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1answer
142 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
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1answer
286 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 ...
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1answer
196 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 ...
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2answers
324 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 ...
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4answers
603 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 ...
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1answer
741 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 ...
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2answers
3k 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 ...
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0answers
3k 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 ...
3
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3answers
864 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
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1answer
418 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 ...
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0answers
198 views

How to estimate the norm of a matrix

There is a matrix as following, ...
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0answers
247 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 ...
10
votes
1answer
859 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
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2answers
629 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 ...