All Questions
Tagged with linear-algebra matrix-analysis
364 questions
4
votes
2
answers
2k
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 ...
- 20.2k
1
vote
0
answers
443
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 $...
- 111
0
votes
0
answers
161
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 ...
- 6,962
2
votes
2
answers
902
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 ...
- 52.3k
2
votes
1
answer
304
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 ...
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- 1
4
votes
3
answers
4k
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?
- 54.2k
6
votes
2
answers
421
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 ...
- 2,239
5
votes
2
answers
562
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 ...
- 6,962
1
vote
2
answers
576
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 ...
- 6,962
1
vote
4
answers
741
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 ...
- 39.9k
3
votes
1
answer
1k
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 ...
- 3,934
1
vote
0
answers
182
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 ...
- 6,962
2
votes
1
answer
851
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 ...
- 20.2k
6
votes
1
answer
737
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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- 1