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.
21
votes
2answers
2k views
What do eigenvalues have to do with pictures?
I am trying to write a program that will perform OCR on a mobile phone, and I recently encountered this article :
Can someone explain this to me ?
32
votes
5answers
7k views
What is the importance of eigenvalues/eigenvectors?
What is the importance of eigenvalues/eigenvectors?
5
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 ...
4
votes
1answer
297 views
eigen decomposition of an interesting matrix
Lets define:
$U=\left \{ u_j\right \} , 1 \leq j\leq N= 2^{L},$ the set of all different binary sequences of length $L$.
$V=\left \{ v_i\right \} , 1 \leq i\leq M=\binom{L}{k}2^{k},$ the set of ...
1
vote
1answer
227 views
eigen decomposition of an interesting matrix (general case)
Lets define:
$U=\left \{ u_j\right \} , 1 \leq j\leq N= b^{L},$ the set of all different sequences of length $L$ where each element of the sequence can be an integer in $\left \{ 0, 1, .., b-1 ...
27
votes
2answers
909 views
Eigenvalue problem: Prove that all of the eigenvalues of $A$ are 1.
Here's a cute problem that was frequently given by the late Herbert Wilf during his talks.
Problem: Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. ...
4
votes
4answers
327 views
Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$
Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues.
If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
2
votes
2answers
912 views
Eigenvalues of product of a matrix and a diagonal matrix
My situation is as follows: I have a symmetric positive semi-definite matrix L (the Laplacian matrix of a graph) and a diagonal matrix S with positive entries $s_i$.
There's plenty of literature on ...
5
votes
4answers
643 views
eigenvalues of certain block matrices
This question inquired about the determinant of this matrix:
$$
\begin{bmatrix}
-\lambda &1 &0 &1 &0 &1 \\
1& -\lambda &1 &0 &1 &0 \\
...
4
votes
1answer
208 views
Characterizing a real symmetric matrix $A$ as $A = XX^T - YY^T$
In my personal research and quest to better understand the subject, I have noticed something concerning the Cholesky factorization of symmetric matrices. Everything I have read states that a symmetric ...
3
votes
2answers
111 views
Matrix proof using norms
I have a linear algebra question I need help with.
Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
3
votes
1answer
216 views
How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?
A few days ago, I had a vague question in my mind about "matrix methods" for finding roots of a polynomial. Now I can ask at least a semi-precise question, thanks to the post
How to calculate complex ...
2
votes
1answer
402 views
Eigenvalues of an operator
I think this question isn't that hard, but I am a bit confused:
Define $$(Af)(x):=\int_{0}^{1}\cos(2\pi(x-y))f(y)dy.$$ Then $A$ is an operator on functions. Find the eigenvalues and the ...
1
vote
1answer
100 views
Hermitian Matrices are Diagonalizable
I am trying to prove that Hermitian Matrices are diagonalizable.
I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are ...
10
votes
2answers
217 views
+150
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:
...
12
votes
1answer
259 views
Eigenvalues for $3\times 3$ stochastic matrices
This is a plot of the non-real eigenvalues of 10000 randomly
generated $3\times3$ stochastic matrices. It's pretty clear
that they lie in the convex hull of the three cube roots of unity.
The ...
6
votes
1answer
855 views
intuition for complex eigenvalues
The eigenvalues of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same.
My question
What is the ...
5
votes
2answers
1k views
Examples for proof of geometric vs. algebraic multiplicity
Here you see a supposedly easy proof of a well-known theorem in linear algebra:
Although I know I should understand this, I don't :-(
Obviously there are too many indices and stuff, so I don't see ...
5
votes
2answers
3k views
Eigenvalues of product of matrices
If $\mathbf{A}_{n\times n}$ is a positive semi-definite matrix with eigenvalues $\{\alpha_k\},\ k\in\{1,...,n\}$, and $\mathbf{B}_{m\times n}$ is an arbitrary matrix with singular values ...
4
votes
2answers
2k views
Eigenvalues for the rank-1 matrix $uv^T$
Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
9
votes
1answer
232 views
Why does this matrix have 3 nonzero distinct eigenvalues
Consider the $n \times n$ matrix $$A=\left[
\begin{array}{cccc}
0 & 1 & ... & 1 \\
1 & 0 & & 0 \\
\vdots & & \ddots & \\
1 & 0 & & 0%
...
2
votes
2answers
428 views
How to prove Weyl’s asymptotic law for the eigenvalues of the Dirichlet Laplacian?
The following comes from Springer Online Reference Works:
Consider a bounded domain $\Omega\subset\mathbb{R}^n$ with a piecewise smooth boundary $\partial\Omega$. $\lambda$ is a Dirichlet eigenvalue ...
11
votes
5answers
3k 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 ...
7
votes
2answers
2k views
Why are all nonzero eigenvalues of the skew-symmetric matrices pure imaginary?
Assume that $A$ is an $n\times n$ skew-symmetric real matrix, i.e.
$$A^T=-A.$$
Since $\det(A-\lambda I)=\det(A^T-\lambda I)$, $A$ and $A^T$ have the same eigenvalues. On the other hand, $A^T$ and ...
6
votes
1answer
463 views
Largest eigenvalue of a positive semi-definite matrix is less than or equal to sum of eigenvalues of its diagonal blocks
This question is very similar to this one.
Let $B$ be a positive semi-definite matrix and $B = \begin{bmatrix} B_{11} & B_{12} \\ B_{12}' & B_{22} \end{bmatrix}$ where $B_{11}$ is $p \times ...
6
votes
4answers
4k 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 ...
2
votes
7answers
2k views
Eigenvalues and eigenvectors in physics?
How can we physically interpret an eigenvalue or an eigenvector in linear algebra?
3
votes
4answers
5k views
Diagonalizable Matrices: How to determine?
I am trying to figure out how to determine the diagonalizability of the following two matrices. For the first matrix
$$\left[\begin{matrix} 0 & 1 & 0\\0 & 0 & 1\\2 & -5 & ...
3
votes
3answers
633 views
Connection between eigenvalues and eigenvectors of a matrix in different bases
If you have a matrix $A$ you can find its eigenvalues and eigenvectors. If you represent this matrix relative to another basis $\mathcal{D}$ you can again find its eigenvectors and eigenvectors.
My ...
3
votes
3answers
716 views
How can I find the nth exponent of the matrix using the diagonalization algorithm?
Hey guys, simple question in linear algebra.
I want to find the nth exponent of this matrix:
$$
\left[
\begin{array}{cc}
1 & 1 \\
0 & 2
\end{array}
\right]
$$
I'm trying to use the ...
2
votes
1answer
145 views
Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$
I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
1
vote
1answer
105 views
Do the non-zero eigenvalues of AB and BA have the same algebraic multiplicity (for AB and BA not square)?
I know that if A and B are square nxn matrices, then AB and BA have the same characteristic polynomial and thus the same eigenvalues (and same algebraïc multiplicity).
I'm wondering though if this ...
1
vote
0answers
62 views
All eigenvalues of matrix $A$ are real, then there exists $B$ such that $B^2=A$ [duplicate]
Possible Duplicate:
Square root of a matrix
Prove that if all eigenvalues of a matrix $A \in \mathcal{M} (n,n; \mathbb{R} )$ are real, then there exists $B \in \mathcal{M} (n,n; \mathbb{R} ...
1
vote
2answers
163 views
Example of eigenvectors in different bases (follow-up question)
This is a follow-up question on this one: Connection between eigenvalues and eigenvectors of a matrix in different bases
Assume I have matrix
$$
B=\left(
\begin{array}{cccc}
0 & 0 & 1 ...
0
votes
1answer
347 views
Basic logic problem with verbal question, confirmation whether right or wrong
Problem 2a here on page 882, translated
Prove the statement
If $\lambda\in \sigma(A)$, so $\lambda^p \in\sigma(A^p) \forall
p\in\mathbb N.$
(where $\lambda$ is an eigen-value and ...
6
votes
2answers
346 views
Eigenvalues of an interesting real symmetric matrix
Problem:
Let A be a square matrix with all diagonal entries equal to 2, all entries directly above or below the main diagonal equal to 1, and all other entries equal to 0. Show that every eigenvalue ...
5
votes
0answers
49 views
What about other symmetric functions of the eigenvalues? [duplicate]
Possible Duplicate:
Identities for other coefficients of the characteristic polynomial
Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
4
votes
1answer
186 views
What are the Eigenvectors of the curl operator?
The curl operator $\vec\nabla\times\mathbb{1}$ can be written as a skew-symmetric 3x3 matrix
$$\mathrm{curl} = \begin{pmatrix}0 & -\partial_z & \partial_y \\ \partial_z & 0 & ...
3
votes
0answers
60 views
Jacobi's Rotation has two possibilities, why do they both result in same upper triangular magnitude norm?
The Jacobi's rotation is the complex Givens rotation (unitary similarity) that results in a zero for a specified element of a matrix. If the element
is not adjacent to the diagonal, then there are ...
3
votes
1answer
51 views
How to be sure that the $k$th largest singular value is at least 1 of a matrix containing a k-by-k identity
In section 8.4 of the report of ID software, it says that the $k$th largest singular value of a $k \times n$ matrix $P$ is at least 1 if some subset of its columns makes up a $k\times k$ identity.
I ...
3
votes
1answer
113 views
Checking if one “special” kind of block matrix is Hurwitz
I have the next block matrix
$$
J = \begin{bmatrix}A & B \\ K &0\end{bmatrix}
$$
all matrices are square, where $A < 0$ (definite negative), $B$ has all its eigenvalues with positive real ...
3
votes
3answers
83 views
Can an eigenvalue (of an $n$ by $n$ matrix A) with algebraic multiplicity $n$ have an eigenspace with fewer than $n$ dimensions?
Is it possible for a matrix with characteristic polynomial $(λ−a)^3$ to have an eigenline (one-dimensional eigenspace)?
I know that geometric multiplicity can generally be smaller than algebraic ...
3
votes
2answers
170 views
When is the geometric multiplicity of an eigenvalue smaller than its algebraic multiplicity?
I was kinda crushed to discover that two different matrices with different properties can actually share the same characteristic polynomial ($-\lambda^3-3\lambda^2+4$):
$A=\begin{pmatrix}
1 & ...
2
votes
5answers
707 views
Are the eigenvectors of a real symmetric matrix always an orthonormal basis without change?
I was reading the wikipedia page for symmetric matrices, and I noticed this part:
...
2
votes
1answer
227 views
How to calculate eigenvectors for equal eigenvalues?
I've had this matrix on a test some time ago:
$$ \begin{bmatrix}
0 & 1 & 0\\
-4& 4 & 0\\
-2& 1 & 2
\end{bmatrix} $$
It's eigenvalues are $\ \lambda _1= \lambda _2= ...
1
vote
0answers
88 views
3 nonzero distinct eigenvalues, part 2
This is an attempt to generalize the answer to a previous question
Consider the $n \times n$ matrix $$A=\left[
\begin{array}{cccc}
0 & \frac{1}{n-1} & ... & \frac{1}{n-1} \\
1 & 0 ...
1
vote
3answers
172 views
Possible eigenvalues of a matrix satisfying certain conditions
This problem appeared in my algebra textbook
Suppose $A$ is a real matrix such that $A^2 = A^t.$ What are the
possible eigenvalues of $A?$
Here is what I tried in order to solve it.
Since the ...
1
vote
2answers
298 views
similar matrices have same eigenvalue… stuck
So $Ax=\lambda x$ where $A$ is the matrix, $\lambda$ and $x$ its eigenvalue and eigenvector resptectively.
Now I have another matrix similar to $A$ i.e. $B=TAT^{-1}$
I'll let a vector $y$ which is ...
1
vote
2answers
436 views
Are singular matrices diagonalizable?
Let $A_{n\times n}$ be Hermitian with eigenvalues $\lambda_1 > \lambda_2 > \ldots > \lambda_r=0$ and multiplicities $q_1,...,qr$. Can $A$ be diagonalized? Is the matrix of eigenvalues
...
1
vote
2answers
149 views
Is there a connection between the diagonalization of a matrix $A$ and that of the product $DA$ with a diagonal matrix $D$?
Given a diagonalizable matrix $A = P_0\Lambda_0 P_0^{-1}$ and a diagonal matrix $D$ with $\det D=1$, is there any connection between $P_0$ and the matrix $P$ of the diagonalization of $DA = P\Lambda ...
