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Results tagged with matrix-theory
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user 13650
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
14
votes
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for ...
Counterexample: Consider the entire function $$ A(z) = \pmatrix{e^z & 0\cr
z & 1\cr}$$
An entire logarithm of $A(z)$ must have eigenvalues $z + 2\pi i n$ and $2 \pi i m$ for …
- 54.2k
11
votes
Accepted
No arbitrary product of matrices has eigenvalue 1?
In the case $n=4$ you could have $D = \pmatrix{0 & 0 & 0 & 1\cr 0 & 0 & 1 & 0\cr
0 & 1 & 0 & 0\cr 1 & 0 & 0 & 0}$, in which case $A_1 A_2 A_3 A_4 A_1 A_2 A_3 A_4 = \pmatrix{0 & 0 & 0 & 0\cr 0 & 0 & 0 …
- 54.2k
8
votes
Differentiability of Eigenvalues - Perturbation Theory
The eigenvalues of a square matrix $A$ are the roots of the characteristic polynomial, and are analytic except where their multiplicities change.
Thus if (in a certain open region of parameter space) …
- 54.2k
7
votes
Accepted
Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.
Your matrix $A = X^T X$ where $X$ is a random $m \times N$ matrix with a continuous distribution having a density. An $m \times m$ submatrix of $A$ is $Q^T A P = (XQ)^T XP$ where $P$ and $Q$ are $N \ …
- 54.2k
6
votes
Accepted
Explicit formula for the functional calculus of 2x2 matrices
A general procedure for $f(A)$ for any $n \times n$ matrix $A$, where $f$ is an analytic function in a neighbourhood of the spectrum of $A$, is this. Let
$p$ be a rational function such that $p(\lamb …
- 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
4
votes
Accepted
when does elementwise-log preserve positive-semidefiniteness?
It's not true that it works for $Z$ small enough. Consider the $2 \times 2$ case
$$ Z = \pmatrix{t & 2t\cr 2t & 4t\cr} $$
which is positive semidefinite for $t \ge 0$.
Then $$\det(X) = \log(1+t)\log( …
- 54.2k
4
votes
Accepted
Can I modify the singular values of a matrix in order to get a negative eigenvalue?
Not necessarily. For example, consider
$$ A = \pmatrix{\cos(\theta) & -\sin(\theta)\cr \sin(\theta) & \cos(\theta)}$$
with eigenvalues $e^{\pm i \theta}$ having positive real part if $-\pi/2 < \thet …
- 54.2k
3
votes
eigenvector update formula
Here is a first-order approximation.
I'll write $\tilde{v} = v + \rho w + O(\rho^2)$ and $\tilde{\lambda} = \lambda + \rho \mu + O(\rho^2)$. We may assume $\|v\| = \|\tilde{v}\| = 1$, so $v^T w = 0$. …
- 54.2k
3
votes
small sums of entries in submatrices - strange phenomenon
Perhaps there's something I don't understand, but your bound is asymptotically best possible.
That is, by taking all $x_i = c$ and the same set for the $d$ rows and the $d$ columns
the sum of entries …
- 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
2
votes
Checking a matrix for distinct rows
The obvious "algebraic" condition is that the left null space of your matrix contains no vector of the form $e_i - e_j$. Does that help?
- 54.2k
2
votes
Algorithm for checking positive definite matrix over a subspace
Of course. Use Gram-Schmidt to construct an orthonormal basis $\{u_i\}$ of $V$, and use your algorithm on the matrix with entries $u_i^\top A u_j$.
- 54.2k
1
vote
Powers of small square matrices over the Laurent polynomial ring with integer coefficients
Your matrix (call it $A(t)$) has characteristic polynomial $\lambda^2 - t \lambda - 1$, so it satisfies $A(t)^2 - t A(t) - I = 0$ and thus
$A(t)^{n+2} = t A(t)^{n+1} + A(t)^n$. For $n \ge 2$ I get
…
- 54.2k
1
vote
power of a block triangular matrix
In general the limit will not exist. For example, the $(2,1)$ block of $M^n$ is $B_n = \sum_{j=1}^n A^{j-1} B A^{n-j}$. By taking a suitable basis, we may assume $A$ is diagonal.
Under the assumption …
- 54.2k