Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with matrix-analysis
Search options answers only
not deleted
user 13650
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
lower bound of a trace quadratic form
I don't know what form of improved bound you could hope for. If $v$ is a vector
in the kernel of $A$, then adding $v v^T$ to $X$ will not change $A X A^T$ but
will increase $\text{tr}(X)$. So if $A …
- 54.2k
1
vote
Maximizing/Minimizing the Operator norms of 0-1 matrices subject to a constraint
In the case $n=2$, all $2 \times 2$ $0-1$ matrices of full rank are equivalent (under permutation of rows or columns) to either $I$ or $\pmatrix{1 & 1\cr 0 & 1\cr}$, so the answer in that case is $\al …
- 54.2k
0
votes
Accepted
Decomposition of one Matrix into six matrices
If $X_1, X_2, X_3$ are invertible, $Y_1$ and $Y_2$ can be any invertible matrices and $Y_3 = (X_1 Y_1 X_2 Y_2 X_3)^{-1} A B$.
- 54.2k
13
votes
Does the matrix exponential preserve the positive-semi-definite ordering?
For example, try $$A = \pmatrix{1 & 0\cr 0 & 0\cr},\ B = A + t C\ \text{where}\ C = \pmatrix{1 & 1\cr 1 & 1\cr}$$
where $t>0$ is small. Then $A ≼ B$ but
$$ e^B - e^A = t \pmatrix{e & e-1 \cr
e …
- 54.2k
2
votes
Accepted
Logarithms of matrices in the disk-algebra
$\Delta(z)$ does not have a nonpositive real eigenvalue for $|z| < 4$, so the principal branch of the logarithm is defined and analytic on a neighbourhood of its spectrum, and thus the holomorphic fu …
- 54.2k
4
votes
Nearby matrices have nearby leading eigenvectors?
The spectral projection of $A$ for eigenvalue $1$ can be realized as
$P_A = \dfrac{1}{2\pi i} \oint_{\Gamma} (z I - A)^{-1} \; dz$
where $\Gamma$ is a contour that encloses $1$ but none of the other …
- 54.2k
4
votes
Accepted
Connection between eigenvalues of A and its LDL decomposition
More generally, the following appears to be true. Suppose $A$ is a real $N \times N$ symmetric matrix with rank $N-1$ and $A b = 0$ where $b \ne 0$.
Let $A = LDL^T$ be the $LDL^T$ decomposition of $ …
- 54.2k
4
votes
Accepted
Bounds on Matrix Exponential
$\|C(k)\|$ can be arbitrarily large, since e.g. you can add some large integer multiple of $2\pi i I$ without changing $e^{C(k)}$. If you want to try to avoid this, you might specify that $C(k)$ is t …
- 54.2k
0
votes
Accepted
Bounding/approximating the largest eigenvalue of the special case of companion matrix
First of all, we can scale your matrix so there is only one parameter, so let's say $p = 1$ for simplicity.
It looks to me like the characteristic polynomial of your matrix is
$$ P(\lambda) = \frac{\ …
- 54.2k
3
votes
Additivity of the Field of Values
This is just a partial answer, but maybe an important case.
If $A$ is hermitian, $F(A)$ is the interval $[\lambda_{\text{min}}(A), \lambda_{\text{max}}(A)]$, where $\lambda_{\text{min}}(A)$ and $\lam …
- 54.2k
10
votes
Induced matrix norm less than one for matrices with spectral radius less than one
Consider the matrix $A = \pmatrix{0 & 1\cr 0 & 0\cr}$ which has spectral radius $0$. But $A A^*$ has spectral radius $1$, so for any sub-multiplicative norm $$1 \le \|A A^*\| \le \|A\| \|A^*\| \le \m …
- 54.2k
3
votes
The space of positive definite orthogonal matrices
A (real) orthogonal matrix $A$ is positive definite if and only the symmetric matrix $M = A + A^T$ is positive definite. There are many equivalent characterizations of this: one is that all leading p …
- 54.2k
11
votes
Accepted
How much redundancy resides in an $n \times n$ orthogonal matrix?
$O(n)$ is a manifold of dimension $n(n-1)/2$, so "generically" one might hope to recover up to $n(n+1)/2$ entries from the other $n(n-1)/2$. This won't quite work, however. Given any set $A$ of rows …
- 54.2k
5
votes
Accepted
The structure of the $n$-th power of a special matrix
The characteristic polynomial of $C_p^{(a,b)}$ is $\lambda^p - (a+b) \lambda^{p-1}$. Therefore, for $m \ge p$ we have $$(C_p^{(a,b)})^m = (a+b)^{m-p} (C_p^{(a,b)})^{p-1}$$
It appears that $B = (C_p^{ …
- 54.2k
3
votes
Accepted
Define a matrix function with a specific property
Why not just take the sum of the norms of the commutators $[H_{ij}, H_{kl}]$ and $[H_{ij}, H^*_{ij}]$?
- 54.2k