Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.

learn more… | top users | synonyms (2)

0
votes
2answers
31 views

Second eigenvector of double eigenvalue matrix

$\begin{bmatrix}\frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2}\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}$ $\lambda_{1} = \lambda_{2} = -1$ ...
0
votes
0answers
10 views

Matrix shear transformations

If you know the line of a shear transformation (the invariant line), how word you go about finding the shear factor? Also, funny as it may sound, what is the shear factor - what does it show?
1
vote
2answers
34 views

Showing that if $A$ is diagonalizable then $A^2-4A+8I$ is diagonalizable

Let $A_{n\times n}$ be a real matrix then: if $A^4 = 8A$ then $A$ is not invertible. if $A$ is diagonalizable over $\mathbb R$ then $A^2-4A+8I$ is diagonalizable over $\mathbb R$. ...
1
vote
1answer
43 views

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way… If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is ...
2
votes
2answers
32 views

Finding Eigenvectors for $3 \times 3$ matrix with rows of zeros.

For a $3 \times 3$ matrix: $ $[A]$ = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} $ I have the eigenvalues: ...
0
votes
1answer
35 views

True/false questions about minimal and characteristic polynomials of a matrix

We have the matrix $A= \begin{pmatrix} 0 &2 &2 \\ 2& 0 &2 \\ 2& 2 & 0 \end{pmatrix}$, then one of the following is true: $f_A(x)=m_A(x) $ The matrix ...
0
votes
0answers
3 views

Not getting dot product of Eigen vectors in MATLAB aslo in LAPACK to zeros

I know that theoretically eigenvectors of real symmetric matrix are orthogonal to each other. So for each pair, dot product will be zero. But when I am calculating eigenvectors from real symmetric ...
2
votes
1answer
26 views

Eigenvalue perturbation theory for $(A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$

Let $A, B$ be $n \times n$ matrices with full rank. I'm interested in getting a bound on how the smallest eigenvalue of $S = (A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$ changes when I perturb $A$ and $B$. ...
0
votes
0answers
15 views

LTI system: solving for the time at which a system state reaches a given value

Suppose I have the following Linear Time Invariant (LTI) system: \begin{equation} \dot{x}(t) = Ax(t) + Bu(t) \end{equation} where $x(t)=\begin{bmatrix}x_1(t) & x_2(t) &\ldots ...
1
vote
0answers
21 views

Repeated Eigenvector/Eigenvalue matrix method

So I am having trouble with finding the generalized solution and I am not sure why my answer is interpreted as incorrect and I wanted to double check. $$ \overrightarrow{y'} = \begin{pmatrix} -6 ...
2
votes
2answers
23 views

Eigenvalues and vectors of a Linear Transformation

I am kinda lost here. All I did until now was finding eigenvalues and vectors for a matrix but as far as I can understand the question it asks me to find the eigenvalues of a Linear Transformation? ...
2
votes
0answers
15 views

Eigenvalues and Eigenvectors of an hyperbolic partial differential equations $\partial_t W + A \partial_x W = 0$

I read in a article dealing with a hyperbolic partial differential equations this statement : For any system of hyperbolic partial differential equations (pde), expressed as (1) ...
0
votes
3answers
42 views

Understanding the method to find Eigenvectors

For a matrix: $ $[A]$ = \begin{bmatrix} 5 & 4 \\ 1 & 2 \\ \end{bmatrix} $ I have the eigenvalues: $\lambda = 6, 1$ Now for each value I need to find ...
2
votes
2answers
54 views

If A and B are diagonalizable then so is AB

When we have to n×n matrices that can be made diagonal (maybe not in the same basis), is it true that the same works for their product?
0
votes
1answer
15 views

Eigenvector with matrix amlost full with zeros

Hi i have weird problem with calculate eigenvector from simplest matrices. So have something like this: $A = \begin{bmatrix} \frac{1}{2} & 0 \\ 2 & \frac{1}{2} \end{bmatrix}$ Eigenvalues are ...
1
vote
0answers
11 views

Posch-Teller potential

I made a matlab code to find numerical solutions to the Schrödinger equation independent of time, and it works very good, the eigenfunctions always are what they should be, the problem is in the ...
2
votes
1answer
38 views

Calculate complex eigenvector

Hi i have problem i hope that someone can make this for me more clear: So i have matrix $A = \begin{bmatrix} -2 & 1 \\ -2 & 0 \\ \end{bmatrix}$ I have to calculate eigenvector as matrix $P$ ...
1
vote
2answers
45 views

condition number of matrix plus constant times identity

I saw this post on the eigenvalues of a matrix plus a constant times the identity matrix. Say $A$ is an $n\times n$ matrix (real and non-singular) with eigenvalues $\lambda_1,\ldots,\lambda_n$, then ...
1
vote
0answers
27 views

rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)

I am reading "A First Course in Partial Differential Equations with Complex Variables and Transform Methods" (Weinberger, p. 168). if we have the eigenvalue problem $$ (pu')'- qu + \lambda \rho u = 0 ...
0
votes
1answer
57 views

An exercise from vector spaces

I do not even know how to start. Find a symmetric matrix $A \in \mathbb R^{3\times 3}$ with the following properties. Both $[1,2,2]^T$ and $[2,1,-2]^T$ are eigenvectors. It has three distinct ...
0
votes
1answer
17 views

Finding null space of symmetric matrix generated by outer product

Let $p, q \in \mathbb{R}^n$ such that $||p|| = ||q|| = 1$ and define $A = pq^T + qp^T$. I am trying to find the null space of $A$, but am not having very much luck. I have managed to show that $p + q$ ...
3
votes
1answer
25 views

Proving that $V = U \oplus W$ where $W$ and $U$ are sets of eigenvectors of $S: V \to V$

Let $V$ be a finite dimensional real vector space, $S : V \to V$ be a linear map such that $S^2 = I$. Show that $V = U \oplus W$ where $U = \{u \in V : Su = u\}$ and $W = \{ w \in V : Sw = -w\}$. ...
0
votes
0answers
11 views

Which numerical method gives the most accurate solutions of Helmholtz equation for arbitrary domains?

There are many numerical methods for the solutions of PDE's such as FDM, FEM, SEM, Meshfree methods etc. I'm wondering which method gives the most accurate Dirichlet eigenvalues (and corresponding ...
0
votes
0answers
12 views

What is the effect on the spectrum by addition of a matrix with that of a rank 2 matrix?

Let $A$ and $B$ be two $n\times n$ matrices with rank of $B$ equal to $2$. Then how is the spectrum of $A$ and $A+B$ related? Or whether we can say something about - which of the eigenvalues of $A$ ...
4
votes
1answer
48 views

Finding a matrix given eigenvalues and eigenvectors.

I am asked to construct a $4 \times 4$ symmetric matrix, with given eigenvalues and eigenvectors. I understand how to actually get $A$ as a product of $P^T, D$ and $P$, when $D$ is the diagonal ...
0
votes
0answers
22 views

What has been already done on spectrum of Hermitian matrices?

Could anyone suggest some books/articles related to the determination of eigenvalues and eigenvectors of some special complex Hermitian matrices?
0
votes
0answers
32 views

Solve system of two homogeneous first-order ordinary differential equa0ti0ns by eigenvectors. (7.16-1)

Please check my work and I shall have a few questions along the way. I am working out of the textbook Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons. The problem to ...
4
votes
3answers
256 views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $e^{X}$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
1
vote
2answers
24 views

If $D$ is a $3\times 3$-matrix of order $6$, then the geometric multiplicity of eigenvalue $-1$ in $D^3$ is two

Consider the subgroup $H \le \operatorname{GL}(3,3)$ generated by the two matrices $$ A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \quad\mbox{ and ...
0
votes
0answers
17 views

Davenport's Q-method (Finding an orientation matching a set of point samples)

I have an initial set of 3D positions that form a shape. After letting them move independently, my goal is to find the best rotation of the original configuration to try to match the current state. ...
0
votes
1answer
43 views

Can we say anything about the relation of the first eigenvectors of a matrix and its rank one updated?

Let $R=S+xx^t$ where $x\in \mathbb{R}^n$ and $R$ and $S$ are $n\times n$ symmetric positive semidefinite matrices. Is there anything I can say about the difference of the first eigenvectors of $R$ ...
0
votes
3answers
47 views

Solutions of $\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&3\\ \end{pmatrix}x=0$

I'm having a bit of confusion here. What are the solutions of $\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&3\\ \end{pmatrix}x=0$ Clearly, $x_2=0$ $x_3=0$ $3x_3=0$ ...
0
votes
1answer
24 views

vector decomposition using eigenvectors

The eigenvalues of the matrix $$ \begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0 \\ \end{pmatrix} $$ are $5$, $0$, and $‐5$. ...
1
vote
0answers
43 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
0
votes
0answers
16 views

For what operators are $sin(ax)$ and $cos(bx)$ eigenfunctions?

So it is clear that the operator $\frac{d^{2}}{dx^{2}}$ is an eigenfunction of $sin(ax)$ and $cos(ax)$. For what other operators are sins and cosines eigenfunctions?
0
votes
3answers
59 views

How to compute the exponential of this given matrix?

I have a problem and can't find any solution. I have the matrix $A= \begin{bmatrix} 0 & 2 \pi \\ -2 \pi & 0 \\ \end{bmatrix}$ and I must compute the matrix $e^A$. I remeber that there was ...
2
votes
0answers
16 views

What can we say about the eigenvalues and diagonalization of this $2N\times2N$ matrix $A$?

There is a $2N\times2N$ matrix $A$, which is of the form: $A=\left(\begin{array}{cc} B & C\\ -C^{*} & -B^{*} \end{array}\right),$ where $B$ is a hermitian matrix, and $C$ is a symmetric ...
0
votes
1answer
33 views

Fundamental set of solutions for First Order Differential Equation

I am a bit unsure about this question and how to approach it and have tried numerous times. It is as follows: $$\overrightarrow{y_1}(t) = \begin{pmatrix} 2e^{3t}-4e^{-t}\\ ...
0
votes
2answers
34 views

Existence of invariant plane for repeated eigenvalues

If we are in $\mathbb{R^3}$ and the characteristic equation has a repeated root, does that always mean that there is an invariant plane rather than just an invariant line, like when there is no ...
-4
votes
1answer
12 views

Eigenvalues of Blockmatrices [closed]

I have a Blockmatrix of the form $ M = \left(\begin{matrix} 0 & 0 \\ A & B \\ \end{matrix}\right) $ $A$ and $B$ are not nessecarily squared, though. Can I conclude that the matrix M has the ...
0
votes
0answers
11 views

Do the generalised eigenvalues of real matrices occur in complex conjugate pairs?

If a real, square matrix $A \in \mathbb{R}^{n \times n}$ has complex eigenvalues, they always occur in complex conjugate pairs. Can the same be said about the generalised eigenvalues, i.e. the ...
0
votes
0answers
23 views

Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. ...
1
vote
0answers
29 views

When does $A$ and $|A|$ have same set of eigenvalues?

Let $A$ be a Hermitian matrix and $|A|$ denotes the entry-wise modulus of $A$. Now what should be the structure of $A$ such that $A$ and $|A|$ will always have the same set of eigenvalues?
0
votes
0answers
17 views

What is the condition under which $\lambda_{\text{max}}=\nu_{\text{max}}$ and $|z|=x$?

Let $A$ be a Hermitian matrix. Let $Az=\lambda_{\text{max}}z; ~z\ne0.$ Consider $|A|$, the matrix with entry-wise modulus of $A$ and let $|A|x=\nu_{\text{max}}x; ~x\ne0.$ Now my question is - under ...
0
votes
0answers
12 views

Is this block matrix Hurwitz?

Let $A\in\mathbb{R}^{n\times n}$ be lower-triangular and Hurwitz. (Hence, the eigenvalues of $A$ are all real and strictly negative). Let $k_1,k_2>0$. Consider the matrix $$ M = \begin{bmatrix} 0 ...
0
votes
1answer
36 views

Eigenvalues and Eigenvectors

For each linear operator $T$ on $V$, find the eigenvalues of $T$ and an ordered basis $\beta$ for $V$ such that $[T]_\beta$ is a diagonal matrix. Where $V = M_{2\times 2}(\mathbb R)$ and $T ...
0
votes
1answer
24 views

Eigenvectors with Irrational elements

After searching through SE, I found this question (Eigenvalues of vectors with irrational entries). This is exactly what I am asking, with a different matrix and different eigenvalues. The problem ...
1
vote
0answers
27 views

Eigenvalue problem $DD^{+}$ (big) vs $D^{+}D$ (small). Spectral relations.

Consider a complex $N\times M$ matrix $D$, in my case $N\gg M$. I have found that: 1) The biggest $M$ eigenvalues of $DD^{+}$ (size $N\times N$) are equal to those of $D^{+}D$ (size $M\times M$). 2) ...
1
vote
0answers
25 views

Explicit example of Gershgorin circle theorem edge case

The Gershgorin Circle Theorem states that if the union of $m$ of the discs is connected, and disjoint from any discs not in the union, then it contains $m$ eigenvalues of the matrix. I am looking for ...
0
votes
1answer
18 views

similar matrices times another matrix

Given $A$ and $B$ are two similar matrices where $A$ is symmetric, thus diagionalizable. Now we know that $A$ and $B$ have the same eigenvalues. I was wondering are $AC$ and $BC$ similar for any ...