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Results tagged with matrix-analysis
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user 1898
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.
3
votes
Accepted
Algorithm for Computing Kronecker's Canonical Form for Matrix Pencils
GUPTRI by Jim Demmel and Bo Kagstrom computes a triangular decomposition that reveals the Kronecker structure of a pencil. It is Fortran code that can be called from Matlab using a Mex-file interface. …
- 20.2k
8
votes
Accepted
How to solve a non-homogeneous quadratic matrix equation?
(commenting about the equation with the plus sign, I hope that the correction was right).
This is one of the few quadratic matrix equations that have a closed form solution. Set $A=-H^{-1}$; then $G …
- 20.2k
2
votes
The eigenvectors and eigenvalues of matrix geometric mean
As for visualization, I have seen SPD 2x2 matrices represented as ellipsoids (such as this one, which would be a 3D case), and the segment joining $A$ and $B$ in that Riemannian distance as a sequence …
- 20.2k
2
votes
How could I extend this result to a case where the matrices were not of full rank?
The natural way to relax the condition $A \geq XB^{-1}X^*$ to the case when $B$ is not full-rank is transforming it to
$$
\begin{bmatrix}
A & X\\
X^* & B
\end{bmatrix} \geq 0,
$$
which is equivalent b …
- 20.2k
8
votes
About Sylvester's determinant
The common ground of those two formulas is related to the Woodbury matrix identity. This relation is a useful statement that shows what happens to the inverse when one "updates" a matrix $A\in\mathbb{ …
- 20.2k
14
votes
Accepted
Are Diagonally dominant Tridiagonal matrices diagonalizable?
Counterexample:
$$
\begin{bmatrix}
-1 & 1 & 0 & 0\\
0 & -1 & 1 & 0\\
0 & 0 & -2 & 2\\
0 & 0 & 2 & -2
\end{bmatrix}
$$
is defective: its eigenvalues are $-1,-1, 0, -4$ (it is block triangular, so its …
- 20.2k
1
vote
Matrix Generator for M/M/1 Queue Waiting Time Distribution
Almost. The generator is
${\bf Q} =\left( \begin{array}{ccccc}
-\lambda & \lambda & 0 & 0 & 0\\
\mu & -(\lambda+\mu) & \lambda & 0 & 0\\
0 & \mu & -(\lambda+\mu) & \lambda & 0 \\
0 & 0 …
- 20.2k
11
votes
Accepted
When the sum of positive definite matrices converges, does the sum of the norm of the associ...
You can bound $\|A_k\| \leq C(n)\max_{i,j} |(A_k)_{ij}|$ for some function of the dimension only $C(n)$, because all norms are equivalent in finite dimension. If I am not mistaken $C(n)=\sqrt{n}$, but …
- 20.2k
2
votes
Woodbury formula
Since there are only few answers, I'll add a shameless plug and advertise one of my papers.
Basically, we first used SMW as a computational tool to speed up a matrix inversion, and then we found out …
- 20.2k
1
vote
Accepted
SVD of two matrices A and B having the same right singular vectors?
Those notes do not refer to the "usual" SVD, but to the generalized SVD, which is a different decomposition of a pair of matrices (and does not require $X$ to be orthogonal, in particular).
For a qui …
- 20.2k
2
votes
Accepted
Reference request: continuity of Cholesky factor
A subtle issue is that $\Pi$ is not unique here. For instance, if
$$
A = \begin{bmatrix}
1 & 0 & 0\\\\
0 & 0 & 0\\\\
0 & 0 & 0
\end{bmatrix}
$$
then you can take both the identity and $(23)$ as the …
- 20.2k
3
votes
How do you solve this quadratic matrix equation?
This question is not an exact duplicate, but my answer gives you a pointer that applies also to your case.
- 20.2k
2
votes
Singularity of matrix pencil-like expression
The statement is false.
Take $3\times 3$ matrices such that $A_{11}=B_{22}=1$ and all other entries are zero. Then $EA-hB$ has the third column equal to $0$, but the row spaces of $A$ and $B$ are disj …
- 20.2k
0
votes
Generate a low-rank sparse covariance matrix
Matlab's sprandsym generates a random sparse positive-definite matrix by starting from a diagonal matrix and applying to it Jacobi rotations, i.e., rotation matrices that act only on two components. I …
- 20.2k
4
votes
What are interesting heuristics of determining how far given matrix is from a singular one?
Another interesting concept in this direction is the pseudospectrum.
(sorry for the one-line answer).
- 20.2k