Number associated to a linear operator from a vector space $V$ to itself: $\lambda$ is and eigenvalue of $T\colon V\to V$ if the map $x\mapsto $\lambda x-Tx$ is not injective.
3
votes
1answer
84 views
Are all Toeplitz matrices diagonalizable?
As in the title. Also, if anyone knows if all Hermitian-symmetric matrices with distinct diagonal elements are diagonalizable, that'd be great to know. Thanks.
Edit: Never mind about the Hermitian ...
3
votes
1answer
67 views
smallest singular value
I know this question is a difficult one, but any advice/tip/reference/heuristic is welcome. Is there any good lower bound (other than $0$) on the smallest singular value of a matrix? It is easy to get ...
3
votes
2answers
272 views
Find the roots of a polynomial using its companion matrix
I would like to find the roots of a polynomial using its companion matrix.
The polynomial is ${p(x) = x^4-10x^2+9}$
The companion matrix $M$ is
$M={\left[
\begin{array}{cccc}
0 & 0 & 0 ...
3
votes
2answers
111 views
Matrix proof using norms
I have a linear algebra question I need help with.
Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
3
votes
1answer
154 views
diagonalization of a bisymmetric matrix
Is there some way to easily diagonalize a rank $n$ bisymmetric Toeplitz matrix with only zeros on its main diagonal? Direct calculation is out of the question, I need some trick... thanks!
Addendum:
...
3
votes
1answer
51 views
How to be sure that the $k$th largest singular value is at least 1 of a matrix containing a k-by-k identity
In section 8.4 of the report of ID software, it says that the $k$th largest singular value of a $k \times n$ matrix $P$ is at least 1 if some subset of its columns makes up a $k\times k$ identity.
I ...
3
votes
1answer
115 views
Checking if one “special” kind of block matrix is Hurwitz
I have the next block matrix
$$
J = \begin{bmatrix}A & B \\ K &0\end{bmatrix}
$$
all matrices are square, where $A < 0$ (definite negative), $B$ has all its eigenvalues with positive real ...
3
votes
2answers
420 views
Lowest Eigenvalue of a positive semi-definite matrix
I am reading a paper on spectral graph theory. Let us say we have an adjacency matrix W and a degree matrix D. We construct a Laplacian matrix, L defined as:
L = D - W
The paper claims that L is ...
3
votes
1answer
172 views
Is this matrix diagonalizable? Wolfram Alpha seems to contradictory itself…
I have the matrix $\begin{bmatrix}0.45 & 0.40 \\ 0.55 & 0.60 \end{bmatrix}$.
I believe $\begin{bmatrix}\frac{10}{17} \\ \frac{55}{68}\end{bmatrix}$ is an eigenvector for this matrix ...
3
votes
1answer
196 views
are there any bounds on the eigenvalues of products of positive-semidefinite matrices?
I have real positive semidefinite matrices (symmetric) $A$ and $B$, both are $n \times n$.
I am looking for upper bounds and lower bounds on the $m$-th largest eigenvalue of $AB$, in terms of the ...
3
votes
1answer
187 views
Physical interpretation: weighted eigenvalues of the Laplacian with a potential
I've already posted this question on Physics.SE, but I thougth it could be useful to ask also here.
No problem if moderators will ask me to cancel this thread... But, please, have mercy! :-D
Let ...
3
votes
2answers
86 views
Comparing the spectrum of the (non)centered matrices
Suppose a symmetric matrix $A\in\mathbb{R}^{n\times n}$ is given. Let $J=I-\frac{1}{n}\cdot 1_n1_n^T\in\mathbb{R}^{n\times n}$ be the centering matrix, with $I$ being the identity matrix, and $1_n=[1 ...
3
votes
1answer
218 views
Eigenvalues of anti-circulant matrices using 1-circulant matrices
Is there any theorem to find the eigenvalues of any anti-circulant matrix using the equivalent (with the same first row) circulant matrix. I found out that,
for any anti-circulant matrix, the ...
3
votes
2answers
212 views
Matrix decomposition in eigenvector product
Is it always possible to decompose an $n$-by-$n$ matrix $\mathbf{A}$ as
$$\mathbf{A} = \sum_{i=1}^n \lambda_i \mathbf{v}_i\mathbf{v}_i^{\rm T}$$
where $\lambda_i$ is the $i$-th eigenvalue of ...
3
votes
2answers
74 views
Diagonalising a $2 \times 2$ and $3 \times 3$ matrix
For each of the following matrices $A$, find an invertible matrix $P$ over $C$ such that
$P^{-1}AP$ is upper triangular:
$$A = \begin{bmatrix}4 & 1\\-1 & 2\end{bmatrix} \quad \text{ ...
3
votes
2answers
47 views
Linear algebra, eigenvectors problem
Suppose you know that A is $2x2$ and symmetric.
Assume the eigenvalues are $4$ and $7$.
An eigenvector for $4$ is the vector $(3, -4)$. What is an eigenvector for $7$?
So first we let ...
3
votes
1answer
33 views
Convergence of eigenvalues $\lambda_{i}^k$
I've been trying to solve this for a while now but can't seem to figure it out. My initial intuition is to just calculate the limit of $\lambda_{2}^k$ using de Moivre's formula like so:
$\lim_{k\to ...
3
votes
1answer
172 views
Evaluating eigenvalues of a product of two positive definite matrices
Let $A,B\in M_n(\mathbb{R})$ be two symmetric positive definite matrices, i.e.: $$\forall x\in\mathbb{R}^n, x\neq 0, (Ax,x)>0, (Bx,x)>0,$$ where $(\cdot,\cdot)$ is the usual scalar product in ...
3
votes
1answer
118 views
Normal operator + only real eigenvalues implies self-adjoint operator?
Let say we are in a complex vector space, is there an example of a normal operator with only real eigenvalues(or without eigenvalues) that is not a self-adjoint operator? Cause of the spectral theorem ...
3
votes
1answer
327 views
Rayleigh-Ritz Theorem
Let $U$ be an $n$-dimensional subspace of $L:=L_2([-1,1])$. Let $F$ be an acting on $L$, given at $f \in L$
$$
(Ff)(x):=\int_{-1}^1 \frac{\sin a(x-y)}{(x-y)}f(y) dy, \quad x \in [-1,1], \quad a>0.
...
3
votes
1answer
147 views
Eigenvalues integral operator - general case
Let $T$ be an integral operator on $L^2([0,1])$, such that:
$$
(Tf)(x) = \int_0^1K(x,y)f(y)dy,
$$
with $K(x,y): [0,1]^2 \rightarrow \mathbb{R}$ continuous and $K(x,y) = K(y,x)$, $K(x,y)\geq0$ $ ...
3
votes
1answer
90 views
Eigenvalues of Hilbert-Schmith operator
I am having trouble determining the eigenvalues and eigenvectors of the operator $Kv(x)= \int_0^1((x+t)v(t)dt$, where the kernel is $k=x+t$. I have tried to solve the equation $Kv(x)=\lambda v(x)$, ...
3
votes
1answer
165 views
A particular Generalized Eigenvalue Problem
Data
I have three $N \times N$ complex hermitian matrices $A=xx^{H}$,$R=rr^{H}$ and a positive-definite matrix $B$. Here $x$ and $r$ are two $N \times 1$ complex vectors. Let $\lambda_{i}, 1\leq ...
3
votes
1answer
147 views
Calculating the inertia of a real symmetric (or tridiagonal) matrix
I'm trying to find a quick method for evaluating the inertia of a real symmetric matrix, though I don't need to evaluate eigenvalues directly. The inertia of a matrix is a triple of the number of ...
3
votes
1answer
730 views
How many possibilties for eigenvectors are there for one eigenvalue?
If I have a $2 \times 2$ matrix $A$, and then I find two eigenvalues $\lambda_1$ and $\lambda_2$ by subtracting $λI$ from A and then taking the determinant=0(singular); to find $\lambda_1$ and ...
3
votes
1answer
154 views
Eigenvalues of a parameter dependent matrix
Given the parameter dependent matrix $A=\begin{pmatrix} 0 & I\\ A_1 & kA_2\end{pmatrix}$, with $A_1, A_2\in \mathbb{R}^{n\times n}$ and $k\in\mathbb{R}>0$, is there a way to display the ...
3
votes
1answer
249 views
Eigenvalues of Self-Adjoint Operator
I'm having difficulties with this homework problem:
Suppose $T$ is a self adjoint linear operator on a vector space $V$. Let $\lambda \in \mathbb{F}$ (where $\mathbb{F}$ is $\mathbb{R}$ or ...
3
votes
1answer
71 views
Eigenvalues of a cyclic symmetric tridiagonal matrix where $M_{k,k+1}=\tfrac12\sqrt{M_{k,k}M_{k+1,k+1}}$
Working on a physics problem, I've encountered some structured cyclic tridiagonal $n\times n$ matrices. They're all of the following form:
$$
\tiny
\begin{bmatrix}
\alpha_1 & ...
3
votes
1answer
229 views
Minimizing the sum of the $k$ smallest elements of the diagonal of a matrix
I have a $N\times N$ symmetric positive semidefinite matrix $Q$, and am considering a class of symmetric positive definite matrices having all eigenvalues in a given bounded interval $[a, b]$.
Is it ...
3
votes
0answers
27 views
If the matrix is positive definite, then its similar matrix is also positive definite?
If $A$ is positive definite and $B$ is similar to $A$.
Can we say that $B$ is also positive definite?
I guess it is true since two matrices have same eigenvalues, and if $\sigma(A) > 0$, and so is ...
3
votes
2answers
67 views
A linear algebra problem
I came across the above problem .Everything looks fine until the last line that says $\longrightarrow$It yields the solution $y_1=(e^t, 0)^t$. I do not know how it came into the picture. A ...
3
votes
0answers
111 views
show that the function satisfies condition of the lemma
Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator
$F$, defined on $L^2([-1,1])$ by
$$
F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
3
votes
0answers
60 views
Jacobi's Rotation has two possibilities, why do they both result in same upper triangular magnitude norm?
The Jacobi's rotation is the complex Givens rotation (unitary similarity) that results in a zero for a specified element of a matrix. If the element
is not adjacent to the diagonal, then there are ...
3
votes
1answer
174 views
How to generate a random matrix whose eigenvalues are less than one
I generate a random matrix. It has this general form:
$$\mathbf{B}=\left[ \matrix{ \mathbf{A}_1&\mathbf{A}_2&\ldots&\mathbf{A}_{p-1}&\mathbf{A}_p \\
...
3
votes
0answers
114 views
Does matrix convergence in $L^p$ imply convergence of the eigenvalues in $L^p$?
Let $A_n(x)$ be a sequence of symmetric matrix functions that converges in $L^p(\Omega)$ to $A(x)$. Is it true that the eigenvalues of $A_n(x)$, or a subsequence of these, converge to the eigenvalues ...
3
votes
0answers
387 views
quick ways to »verify« determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …
What are easy and quick ways to »verify« determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after caculating them?
So if I calculated determinant, minimal ...
3
votes
1answer
401 views
Relationship between ellipsoid radii and eigenvalues
I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup.
I went through a theorem in the book stating the ...
3
votes
0answers
399 views
Numerical methods to find eigenvectors with 0 eigenvalue
I'm curious if there's any numerical way of directly finding the eigenvectors with eigenvalue 0.
If I didn't have to do it directly, I would probably do it like this in pseudocode:
...
2
votes
7answers
2k views
Eigenvalues and eigenvectors in physics?
How can we physically interpret an eigenvalue or an eigenvector in linear algebra?
2
votes
3answers
2k views
Do real matrices always have real eigenvalues?
I was trying to show that orthogonal matrices have eigenvalues 1 or -1.
Let $u$ be an eigenvector of $A$ (orthogonal) corresponding to eigenvalue $\lambda$. Since orthogonal matrices preserve length, ...
2
votes
3answers
308 views
How can I quickly calculate the eigenvalue and eigenvector for this matrix?
I wonder if there is any trick to calculate the eigenvalue and eigenvectors for the all-1 matrix, namely
$A=%
\begin{bmatrix}
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & ...
2
votes
5answers
714 views
Are the eigenvectors of a real symmetric matrix always an orthonormal basis without change?
I was reading the wikipedia page for symmetric matrices, and I noticed this part:
...
2
votes
4answers
152 views
Determinant of matrix exponential?
Suppose $A$ is a $n \times n$ constant matrix. How can I prove
$\det(e^A) = e^{\displaystyle \sum_{\lambda_i\in\sigma(A)} \lambda_i}$,
where $\sigma(A)$ is the multiset of eigenvalues of $A$?
The ...
2
votes
2answers
1k views
Linear Algebra, proof about eigenvalues
I've been trying to complete a proof for a while now, but I can't. It would be great if someone could finish it for me so that I can at least learn from the solution.
The problem is asked like this:
...
2
votes
3answers
213 views
Eigenvalues of operators
I have a linear operator $T$ which acts on the vector space of square $N\times N$ matrices in this way:
$T(A)=0.5(A-A^\mathrm{t})$
($A^\mathrm{t}$: the transpose of a matrix $A$).
I need to prove ...
2
votes
1answer
187 views
Finding eigenvalues of matrix of matrix.
Let A be 4 X 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following option is an eigenvalue of $\begin{bmatrix}A & I\\I & A\end{bmatrix}$, where I is 4 X 4 identity matrix?
...
2
votes
2answers
585 views
Minimal polynomials and characteristic polynomials
I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices.
When are the minimal polynomial and characteristic polynomial the ...
2
votes
1answer
139 views
Let $B$ be a nilpotent $n\times n$ matrix with complex entries let $A = B-I$ then find $\det(A)$
Let $B$ be a given nilpotent $n\times n$ matrix with complex entries. Let $A = B-I$ find out $\det(A)$. What if B is orthogonal or skew symmetric matrix? Then can we say anything about its trace and ...
2
votes
2answers
63 views
Minimum eigenvalue and singular value of a square matrix
How to show that the relationship $\left | \lambda_{min} \right | \geq \sigma_{min}$ holds between the minimum eigenvalue and singular value of a square matrix $A \in \mathbb{C}^{n \times n}$?
2
votes
3answers
71 views
Linear Algebra Question on Eigenvalues
I am having a difficult time with the following question. Any help will be much appreciated.
Let $A$ be an $n×n$ real matrix such that $A^T = A$. We call such matrices “symmetric.” Prove that the ...
