Tagged Questions
16
votes
1answer
623 views
Ratio of largest eigenvalue to sum of eigenvalues — where to read about it?
Let $E_j$ be the $j$th largest-magnitude eigenvalue of a real symmetric $N \times N$ matrix $M$. I've found that the ratio
$$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}},$$
is a measure of the "rank-one-ness" ...
11
votes
5answers
4k views
Matrix Inverses and Eigenvalues
I was working on this problem here below, but seem to not know a precise or clean way to show the proof to this question below. I had about a few ways of doing it, but the statements/operations were ...
11
votes
2answers
544 views
Why are the eigenvalues of these “bitwise XOR matrices” integers?
In the course of playing around with this question, I have hit upon a question of my own.
Consider the $n\times n$ symmetric matrix $\mathbf X$ whose entries are given by ...
11
votes
3answers
368 views
Eigenvalues of some peculiar matrices
While I was toying around with matrices, I chanced upon a family of tridiagonal matrices $M_n$ that take the following form: the superdiagonal entries are all $1$'s, the diagonal entries take the form ...
10
votes
5answers
2k views
Relation between Cholesky and SVD
When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed).
Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
10
votes
2answers
290 views
A matrix w/integer eigenvalues and trigonometric identity
Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated:
Let $n$ be a natural number.
(a) Consider the following Toeplitz/circulant symmetric matrix:
...
10
votes
1answer
182 views
Eigenvalues of $A$ and $A + A^T$
This question has popped up at me several times in my research in differential equations and other areas:
Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related?
...
8
votes
4answers
259 views
What is the fastest way to find the characteristic polynomial of a matrix?
Finding the characteristic polynomial of a matrix of order $n$ is a tedious and boring task for $n > 2$.
I know that:
the coefficient of $\lambda^n$ is $(-1)^n$,
the coefficient of ...
8
votes
5answers
209 views
Matrices with eigenvalues 0 and 1
How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1?
My attempt:
I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
8
votes
1answer
284 views
How to diagonalize this matrix?
Consider the $n\times m$ matrix $M=[M_1, \ldots, M_m]$ where the $i$-th column reads
$$
M_i= \,^t(\underbrace{1,\ldots,1}_{a_i},0,\ldots,0)
$$
where the $a_i$'s are given positive natural numbers.
...
7
votes
2answers
5k views
Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks
Let $A$ be a block upper triangular matrix:
$A = \left( \begin{matrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{matrix} \right)$ where $A_{1,1} ∈ C^{p×p}$, $A_{2,2} ∈ C^{(n-p)×(n-p)}$
Show ...
6
votes
4answers
5k views
Similar matrices have the same eigenvalues with the same geometric multiplicity
Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities.
Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices ...
6
votes
3answers
1k views
Eigenvalues and Eigenvectors of $2 \times 2$ Matrix
Let's say I have a $2 \times 2$ matrix (actually the structure tensor of a discrete image - I):
$$ \begin{bmatrix}
\frac{\partial I}{\partial x}\frac{\partial I}{\partial x} & \frac{\partial ...
6
votes
1answer
2k views
Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)
First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an ...
6
votes
3answers
143 views
Determine whether a $3\times3$ matrix has a positive eigenvalue?
Given a $3\times3$ matrix is there a criterion capable of telling whether the matrix has a positive eigenvalue?
