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Results tagged with matrix-analysis
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user 11142
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
1
vote
How can I find minimum and maximum eigenvalue of non-positive define matrix
Find your greatest in absolute value eigenvalue, call it $E.$ If it is positive, the matrix $M + 2 EI$ is positive definite, so do whatever you do for positive definite matrices. If it is negative, $M …
- 96.4k
5
votes
Accepted
Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues
A Hermitian matrix of rank $r$ can be represented uniquely as $U D U^\ast,$ where $U$ is an $n\times r$ matrix with orthogonal rows of unit length, and $D$ is an $r\times r$ matrix (this is the singul …
- 96.4k
10
votes
Trace of non-commutable matrices
Not to take anything away from Suvrit's answer, but this is actually much simpler. First, we can assume $M_1$ is diagonal. Call it $diag(x_1, \dotsc, x_i).$
Then the difference between the LHS and the …
- 96.4k
3
votes
complexity of computing the singular vector corresponding to the smallest singular value
The question has been studied at some length. See, for example,
Hubert Schwetlick and Uwe Schnabel, MR 1997360 Iterative computation of the smallest singular value and the corresponding singular vec …
- 96.4k
1
vote
finding a unitary submatrix inside a random matrix
(1) a.s yes
(2) The probability is $0.$ Unitary submatrices are of positive codimension in $GL(r),$ and since there only a finite number of $r\times r$ submatrices, the probability is $0.$ If you want …
- 96.4k
2
votes
Finding matrices $A$ such that the entries of $A^n$ have specified signs
To follow up on Terry's comment (actually, I did the computation before seeing it, but whatever): in the parabolic case, where the matrix has the form
$$A = \begin{pmatrix} d & -b \\ -c & a\end{pmatr …
- 96.4k
2
votes
Accepted
How to determine an unitary operator involved in an unitary transformation?
An algorithm for arbitrary matrices is given by Heydar Radjavi in 1962 (On unitary equivalence of arbitrary matrices, TAMS). The "arbitrary" in the title is there because the problem is trivial for no …
- 96.4k
1
vote
Counting Boolean Normal Matrices of size $2n \times 2n$
Since the dimension of the variety of normal matrices is the same as that of the variety of symmetric matrices (for $n\times n$ complex normal matrices, the real dimension of the variety is $n^2+n,$ s …
- 96.4k
2
votes
Is $1/\max(i,j)$ a bounded matrix on Hilbert spaces?
I could be very confused (likely), but notice that your matrix the sum of $N$ and $N^t,$ where $N$ is the upper triangular matrix where $N_{ij}= 1/i,$ when $i<j$ and $0$ otherwise.
It seems that by D …
- 96.4k
1
vote
Accepted
Equivalent metrics on symmetric positive definite matrices
Check out:
Reverse inequality to Golden–Thompson type inequalities: Comparison of $e^{A+B}$ and $e^Ae^B$
Jean-Christophe Bourin, Yuki Seo (2007), Linear Algebra and its Applications
Volume 426, Issue …
- 96.4k
2
votes
Spectrum of transition matrix for symmetric random walk
The spectrum of the matrix is computed in the beginning of my preprint:
Rivin, Igor. "Growth in free groups (and other stories)." arXiv preprint math/9911076 (1999).
(there is a published version, …
- 96.4k
1
vote
Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.
There is a very nice set of notes by Roman Vershynin...
- 96.4k
2
votes
Accepted
can eigenvector be found without computing the eigenvalue
If you pick a random vector $v$ and look at $v_n=A^n v/\| A^n v\|,$ that will converge to the dominant eigenvector.
- 96.4k
0
votes
Nonlinear eigenvalue problem - sorta
This does not answer the question as stated, but your specific question (assuming the inverse is coordinate-wise) can be stated as
$\min_i \min x_i,$ subject to the system of constraints
$x_i <A_i, …
- 96.4k
4
votes
Exponentiating 4 by 4 matrix analytically
There is a completely explicit formula in this paper of Bensauod and Mouline (rendicotti Palermo, 2005), which is quite compact for low dimensions.
- 96.4k