All Questions
7 questions
13
votes
3
answers
2k
views
Eigenvalue pattern
We consider a matrix
$$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$
One easily ...
- 133
12
votes
6
answers
2k
views
Differentiability of eigenvalues of positive-definite symmetric matrices
Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda_1,\cdots,\lambda_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall ...
- 3,423
11
votes
1
answer
928
views
Imaginary eigenvalues
Consider the matrix
$$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$
This matrix is ...
- 124
3
votes
1
answer
791
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
1
vote
1
answer
2k
views
Positive matrix and diagonally dominant
There is a well-known criterion to check whether a matrix is positive definite which asks to check that a matrix $A$ is
a) hermitian
b) has only positive diagonal entries and
c) is diagonally ...
- 183
1
vote
0
answers
266
views
Interlacing Suleĭmanova spectra
A set of real numbers $\{\lambda_1, \dots, \lambda_n \}$, $n \geq 1$, is called a Suleĭmanova spectrum if it contains exactly one positive value and $\sum_{i=1}^n \lambda_i \geq 0$. (It is well-known ...
- 1,719
0
votes
1
answer
262
views
Perturbing a normal matrix
Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\...
user136156