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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.
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
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
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
3
votes
Triangularizing a matrix with function entries
I am pretty sure this question (actually, the Jordan canonical form question) is studied at great length in Kato's perturbation theory for linear operators book (chapter 1). Kato is also very careful …
- 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
4
votes
Accepted
Spectral properties of the LDL^T matrix factorization
Well, for the closely related Cholesky factorization, there is the following:
Fast Accurate Eigenvalue Computations Using the Cholesky Factorization (1997) (by Roy Matthias), which says that the eige …
- 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
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
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
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
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
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
9
votes
The multiplicity of the max eigenvalue in matrix multiplication
I am afraid you can infer almost nothing. For example, it is a result of Frobenius (1910) that every square matrix is a product of two symmetric matrices. Since the eigenvalues of the symmetric matric …
- 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