2
votes
1answer
45 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
0
votes
0answers
14 views

Finding characteristic roots and characteristic vectors

V is a two-dimensional vector space over the field of real numbers, with a basis $v_1, v_2$. Find the characteristic roots and corresponding characteristic vectors for T defined by $v_1(T) = v_1 + ...
2
votes
2answers
33 views

Definition: Eigenvalues of a matrix

1) Can a non-square matrix have eigenvalues? Why? 2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible. Thank you!
0
votes
1answer
30 views

Matrix multiplication and rank reduction? - What is the minimal polynomial?

given is a matrix A with $\begin{pmatrix} a & 1 & 0 & \cdots & 0 \\ 0 & a & 1 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 ...
1
vote
1answer
18 views

Prove that adjacency matrix has negative eigenvalue

We are given non-oriented graph without loops. Task is to prove that adjacency matrix of that graph has negative eigenvalue. I put some effort into drawing a proof here , but it seems that I'm ...
1
vote
1answer
25 views

Does this question make any sense - eigenvalues and norms

Im having difficulties understanding this question: show that if $b$ is an eigenvector of an invertible matrix $A$ with an eigenvalue $\lambda_1$ and $\delta b$ is an eigenvector of $A$ with an ...
0
votes
2answers
38 views

Eigenvalues of negative companion matrix

Here's a homework question I've been stuck on for a while. Given $A = \left[ \begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & a_0 \\ -1 & 0 & 0 & \cdots & 0 & ...
2
votes
1answer
80 views

$AB=BA$. Prove $B$ is diagonalizable.

Question: A and B are matrices size $n\times n$ given $AB=BA$ and A has n eigenvalues, prove $B$ is diagonalizable. I would have written what I tried to do, but It's really nothing worth reading.. ...
1
vote
2answers
74 views

Are the eigenvalues of $AB$ the same as the eigenvalues of $A$ if $B$ is invertible?

Question: Let $\lambda$ be an eigenvalue of $AB$; $B$ is invertible. Is $\lambda$ also an eigenvalue of A? My thoughts: I think it can happen iff $B$ is identity matrix. If $B$ is the identity ...
0
votes
1answer
29 views

Find the eigenvalue of $\begin{bmatrix}6&-3\\-3&6\end{bmatrix}x=\frac{\lambda}{18}\begin{bmatrix}4&1\\1&4\end{bmatrix}x$

Find the eigenvalue of $\begin{bmatrix}6&-3\\-3&6\end{bmatrix}x=\frac{\lambda}{18}\begin{bmatrix}4&1\\1&4\end{bmatrix}x$ ...
5
votes
3answers
63 views

A question on eigenvalues

Let $A,B\in M_{2}(\mathbb{R})$ so that $A^2 = B^2 = I$. Which are eigenvalues of $AB$? 1) $1\pm \sqrt 3$ 2)$3 \pm 2\sqrt2$ 3)$\dfrac {1}{2},2$ 4)$2 \pm 2\sqrt 3$
4
votes
3answers
43 views

Show non-symmetric matrix has non-orthogonal eigenvectors

I'm struggling with a problem from Boas's Mathematical Methods in the Physical Sciences. The question is, for a 2x2 matrix M s.t. M is real, not symmetric, with eigenvalues real and not equal, show ...
0
votes
1answer
67 views

Systems of linear differential equations - eigenvectors

Solve the following system of equations $ \begin{cases} x_1^{'}(t)=x_1(t)+3x_2(t) \\ x_2^{'}(t)=3x_1(t)-2x_2(t)-x_3(t) \\ x_3^{'}=-x_2(t)+x_3(t)\end{cases} $. First, I create the column vectors ...
1
vote
2answers
39 views

Eigenvalues and eigenvectors of an operator

I have $Ku(t)=\int_0^1 G(t,s) u(s) ds, u\in L^2[0,1]$ where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq 1\end{cases}$$if the eingenvalues of $K$ are $1/k^2\pi^2$ ...
0
votes
1answer
38 views

Eigenvalues and Corresponding Eigenspace Bases

Could someone describe the eigenvalues of $ \left( \begin{array}{cc} 2 & 1 \\ -1 & 2 \end{array} \right) $, as well as the bases of the corresponding eigenspaces? I received eigenvalues ...
0
votes
1answer
34 views

Stochastic matrix problem

A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to 1. Let $A$ be any (general) 2x2 stochastic matrix. a) Show that one of the ...
1
vote
4answers
84 views

Algebra-sum of entries in each column of a sqaure matrix = constant

This is a question from an algebra homework and I am just looking for some tips. The question is: We have: $M$: an $n\times n$ matrix with real entries $c$: a real constant the ...
1
vote
2answers
49 views

Finding Eigenvectors of 3 x 3 matrix

Yes, I have searched for this question. I didn't see another question that was asking the same thing, and the answers seemed to gloss over what I'm missing. I know this is really simple but I can't ...
1
vote
3answers
113 views

Find all $2\times2$ matrices that have pure imaginary eigenvalues.

Find all $2 \times 2$ matrices that have pure imaginary eigenvalues. That is, determine conditions on the entries of a matrix that guarantee the matrix has pure imaginary eigenvalues. ...
2
votes
0answers
53 views

Compute eigenvector using given eigenvalue

I have a fairly small matrix (cca 100x100). I have computed first four largest eigenvalues (with modified Lanczsos and Lanczsos with reorthogonalization). I know this might be a rather dumb question, ...
1
vote
1answer
98 views

Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
1
vote
3answers
62 views

Orthogonal eigenvectors of the matrix

Whether there is a variable $\alpha$ such that the eigenvectors of the matrix $A$ are orthogonal? $A=\begin{bmatrix} 3&1&1\\0&2&1\\0&\alpha&4\end{bmatrix}$
2
votes
1answer
78 views

Are $A,B$ similar matrices? Check my proof

We are given $A,B$ are orthogonal $4$ by $4$ matrices with real values only. We are given $\det(A) = \det(B) = 1$ and $\mathrm{trace}(A) = \mathrm{trace}(B)$. Is $A$ similar to $B$? My solution: I ...
1
vote
1answer
32 views

Eigenvalue of transition matrix

Background: This is an exam problem which I was not able to solve entirely. After the exam, I discussed with other students, almost in vain. The problem: Let $V$ be a $n$-dimensional linear space ...
1
vote
3answers
115 views

Diagonalizable matrices that commute share eigenspace

I know it's been answered before (at least to the case with $n$ different eigenvalues) but I didn't find a proof for the general case, and I would like some help with this question. We are given ...
2
votes
1answer
163 views

prove that if $\lambda$ is an eigenvalue of T then $\bar\lambda$ is eigenvalue of $T^*$

I have to prove that if $\lambda$ is an eigenvalue of T then $\bar\lambda$ is eigenvalue of $T^*$ (adjoint) I know that $<Tv,u> = <\lambda v,u> = ...
1
vote
1answer
77 views

Repeated Iteration of a 2x2 matrix

Suppose I am given a $2$x$2$ matrix $A=$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} And an initial vector $x_n$ = \begin{pmatrix} x_0 \\ y_0\end{pmatrix}. Under repeated iteration $x_{n+1} ...
1
vote
1answer
80 views

Jordan Normal form: problems with finding correct generalized eigenvectors

I have been tasked to find the Jordan Normal Form for the matrix $A$ shown below. \begin{align*} A = \begin{pmatrix} 2 & 2 & 0 & -1 \\ 0 & 0 & 0 & 1 \\ 1 & 5 & 2 & ...
1
vote
3answers
187 views

How to put $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ in canonical form

We are given the equation $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ We did an example of this in class but the equation had less terms. I took a note in class that says : if there are linear terms, I have ...
0
votes
1answer
21 views

Finding eigenvalues for linear map…is this the correct approach?

Let $A= \begin{pmatrix} a_{11}&a_{12}\\a_{21}&a_{22}\\ \end{pmatrix}$ $\in M_2(\mathbb R)$. Define a map $H_A :M_2(\mathbb R) \rightarrow M_2(\mathbb R)$ by $$H_A(B)= \begin{pmatrix} ...
2
votes
1answer
139 views

Finding the eigen vectors of a 3x3 matrix

We are given the matrix: $$A = \begin{bmatrix}1 & 2 & 0\\2 & 1 & 0\\0 & 0 & 1\\\end{bmatrix}$$ I want to find the eigenvalues I did so by solving $$|\lambda I - A| = 0 $$ ...
0
votes
1answer
34 views

A question about the eigenvector and the basis

Let $(1, 0, 0)^T$ and $(0, -1, 1)^T$ be eigenvector of a 3x3 matrix $A$ with eigenvalue 1 and $(-2, -2, 1)^T$ be an eigenvector of $A$ with eigenvalue 2.Put $e_3=(0, 0, 1)^T$. Find eigenvector $v$ ...
1
vote
1answer
36 views

Eigenvector and Its Span

Let $V$ be a vector space over the field $F$ and let $T$ be a linear transformation from $V$ to $V$. Let $v\in V$ such that $v\neq 0$, let $W=span\{v\}$. Prove that if $T(W) \subset W$, then $v$ is an ...
4
votes
1answer
94 views

Find characteristic polynomial of $\,A^2$ if the characteristic polynomial of $\,A$ is $\,t^4 -t$

$A \in M_{4\times4}(\mathbb{R})$ . The characteristic polynomial of A is $P_A(t)=t^4-t$. I have to find the characteristic polynomial of $A^2$ and $A^4$ So I know that due to the Cayley–Hamilton ...
1
vote
1answer
56 views

Prove for a $7\times7$ matrix that the set of all eigenvectors is linearly independent.

Suppose $A$ is a $7\times7$ matrix, $\left\{\vec{v_{1}},\vec{v_{2}}\right\}$ is a basis for $\operatorname{Eig}(A,3)$, $\left\{\vec{v_{3}},\vec{v_{4}},\vec{v_{5}}\right\}$ is a basis for ...
1
vote
2answers
86 views

Prove that $\vec{v}$ is also an eigenvector for $A^{k}$(k = a positive integer). What is the corresponding eigenvalue?

Prove that $\vec{v}$ is also an eigenvector for $A^{k}$(k = a positive integer). What is the corresponding eigenvalue? What I have started with is, $A=(CDC^{-1})$ which can be used to prove ...
1
vote
3answers
116 views

If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?

If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue? I don't really know where to start with this one. I know that ...
1
vote
1answer
92 views

On max-min representation for the principal eigenvalue of second order elliptic operator

(Just to be upfront about things, this is a homework problem.) I'm asked to show that the principal eigenvalue, $\lambda_1$ of an uniformly elliptic operator can be represented by \begin{equation} ...
0
votes
1answer
65 views

Eigenvalue problem?

"Solve the eigenvalue problem or show that it has no solution: $y'' + 2y = x$ for $y(0) = y(\pi) = 0$" I have managed to find a solution to the boundary value problem by finding the complementary ...
0
votes
1answer
40 views

Generalized minimal residuals: eigenvalues and sets of functions

Can someone help me on this exercise (2 parts)? Thanks! Suppose that $S \subseteq \mathbb{C}$ is a set whose convex hull contains $0$ in it's interior (so $S$ is contained in no half-plane ...
0
votes
1answer
75 views

On eigenvalues, hermitian matrices and SVD

Are my ideas on the following "true or false"-statements correct? If $A$ is hermitian and $\lambda$ is an eigenvalue of $A$, then $|\lambda|$ is a singular value of $A$. My answer would be ...
1
vote
0answers
87 views

eigenvalues with strictly negative real parts

$\textbf{Question: }$ If all the eigenvalues $\lambda_i$ of an $n\times n$ matrix $A$, have a strictly negative real part then prove that all the coefficients $a_j$ of the characteristic polynomial ...
2
votes
2answers
242 views

Geometric means: Eigenvalue, eigenvector

Find the eigenpairs for the matrix $M=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$ in terms of $\theta$. It seems that the eigenvalues are $e^{±i\theta}$ ...
0
votes
1answer
120 views

Integral operator

The space $C$ of continuous functions $f(u)$ on the interval $[0, 1]$ is one of many infinite-dimensional analogues of $\mathbb{R}^n$ , and continuous functions $A(u, v)$ on the square $0 \leq u, v ...
3
votes
2answers
1k views

Linear Algebra: Distance between two parallel lines

Find the distance between the two (obviously parallel) lines below where $\alpha ,\beta \in \mathbb R$ are scalars. $$\text{Line ...
1
vote
1answer
154 views

Linear Algebra : Eigenvalues and rank

1) A $4\times4$ square matrix has distinct eigenvalues $\{0, 1, 2, 3\}$. What is its rank? 2) Let $a,b\in\mathbb{R}^n$ be two non-zero linearly independent vectors, and let ...
1
vote
2answers
44 views

need help on proof question on matrices MEI FP2

Matrix M is (n × n). For n=2 and n=3 prove that if the sum of the elements in each row of M is 1, then 1 is an eigenvalue of M. I know that to find eigenvalues and corresponding eigenvectors, the ...
1
vote
0answers
43 views

Casimir Invariants within the universal enveloping algebra

I've been asked to determine the eigenvalue of the Casimir invariant $I_2$ on any irreducible module with highest weight $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$, where; $$I_m = ...
0
votes
2answers
43 views

proofreading for positive definite matrix has positive eigenvalues

$$\vec{v}^{t}\textbf{A}\vec{v} > \textbf{0}\text{ and }\textbf{A}\vec{v} = \lambda\vec{v}\quad \Rightarrow \lambda>\textbf{0}\quad(\mathbb{F}=\mathbb{R}) $$ proof: ...
0
votes
2answers
78 views

Eigenvalue, linear mapping, matrix representation

Suppost that $A=(\vec v_1 \dots \vec v_n)$ is a basis for a vector space $V$ over $F$ and that the \vec v_i are eigenvectors for a linear mapping $T:V-> V$ Meaning that for each 1<=i<= $n$ ...