1
vote
3answers
47 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
-1
votes
2answers
47 views

Matrix with eigen values given find [closed]

Let$$ P=\begin{bmatrix} 0&-2&-3 \\ -1&1 &-1 \\ a&2 &b \end{bmatrix},$$ for some $a,b \in \mathbb{R}.$ Suppose that $1$ and $2$ are eigenvalues of $P$ and $$ ...
0
votes
2answers
30 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
1
vote
1answer
32 views

What's the connection between rank of matrix and $0$ eigenvalue?

My matrix B is nxn and know nothing about if diagonalizble, but I know that rank B = 1. Therefore the geometric multiplicity of λ=0 as an eigenvalue is n-1. But by knowing the rank is 1, can I say ...
2
votes
1answer
22 views

$U\in{Mat(\mathbb{C})} , $$U$ is unitary and $U+iI$ is self adjoint, prove: $U = -iI$

I'm preparing for a test and this question was a real pain to do. I get a bit confused by all the terms, so I'd appreciate if you guys peek at my attempt to solve this, and see if its correct. What I ...
3
votes
1answer
22 views

Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$ Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix) Prove: $A$ is ...
0
votes
1answer
40 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
2
votes
1answer
81 views

Proof that a is an eigen value of p(T) if and only if a=p(lambda) for some eigenvalue lambda of T

$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\C}{\mathbb{C}} \newcommand{\LM}{\mathcal{L}}$ Question: Suppose $\F = \C, T \in \LM(V), p \in ...
0
votes
2answers
43 views

Eigenvaluse and eigenvectors of differential equation

Vector space $C_2$ consists of every function that's second derivative is continuous in [0,1]. Tensor $A$ on the space $C_2$ is defined as $Ay=\frac {d^2 y}{dt^2}$ where $y(t)$ is a vector on space ...
1
vote
4answers
367 views

Linear algebra, power of matrices

$P^{-1}AP = \begin{pmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 2 \\ \end{pmatrix} $ with $P= ...
0
votes
1answer
30 views

Find the eigenvalues of…

My characteristic equation starts off: $$\lambda(\lambda(\lambda-3k)+3k^2)-k^3=0$$ Once expanded I get: $$\lambda^3-3\lambda^2k+3\lambda k^2-k^3=0$$ Where do I go from here?
0
votes
0answers
26 views

Sum of (complex) eigenvalues of a matrix over the rationals

I have a matrix $A \in M_{nxn}(\mathbb{Q})$ and have to show that the sum of the (not necessarily pairwise different) eigenvalues $\lambda_1, \lambda_2, ..., \lambda_n$ of this matrix is rational. ...
1
vote
1answer
32 views

Proof: If $F^3 = F$ then F is diagonalisable

let $V$ be a $\mathbb{R}$-vectorspace with $dim V < \infty$ and $F$ an endomorphism of V with $F^3 = F$. Show: F is diagonalisable. $F^3 = F$ is equivalent to $F^3 - F = 0$. Now I know that ...
2
votes
2answers
41 views

Gershgorin discs and norm of a matrix

Find a matrix, where the estimation of eigenvalues with the help of Gershgorin discs is a, the same as b, worse as the estimation with the help of the norm of the matrix ($||A||_\infty$) So, yes, ...
7
votes
3answers
275 views

What's the Jordan canonical form of this matrix?

given is the $6 \times 6$-matrix $A$: $A = \begin{pmatrix} 0 & 1 & 0 & -1 & 0 & 0 \\ 0 &0&1&1&-1&0\\ -1&0&0&0&-1&-1 \\ 1 & ...
0
votes
1answer
33 views

Eigenvector when all terms in that column are zero?

so I have this matrix: $$ \begin{matrix} 0.7 & 0 & 0 \\ 0.1 & 0.6 & 0 \\ 0 & 0.2 & 0.8 \\ \end{matrix} $$ I managed to solve ...
0
votes
1answer
61 views

Invariant subspaces for endomorphisms with associated Jordan matrices

I would like to know which are the invariant subspaces for the endomorphisms $f1$, $f2$, $f3$, $f4$, $f5$ from vector space $V$ that have the next associated Jordan matrices: $J1 = \left( ...
0
votes
1answer
75 views

Power method for finding all eigenvectors

This is my homework. I was asked to find all eigenvectors of a symmetric and positive definite matrix by inverse power method with shifted. I encountered three problems: The eigenvalues to the ...
1
vote
0answers
37 views

Why every $n \times n$ matrix $A$ has $n$ linearly independent generalized eigenvalues?

Why every $n \times n$ matrix $A$ has $n$ linearly independent generalized eigenvalues? I don't understand the theorem, Can somebody help me prove it?
1
vote
1answer
65 views

Finding Eigenvalue for cubic equation

I'm learning finding eigenvalues. I learned how to find simplistic eigenvalues for $3\times3$ matrix. By using below way. With this way I can only solve if I have simple determinant equation, like ...
0
votes
0answers
35 views

What does my teacher mean by 'choosing' from a vector?

I'm revising some lecture notes from a class I missed, I'm just struggling to figure out what she means at this point. What is choosing x1=0, x2=1... etc mean? Could someone explain?
2
votes
1answer
58 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 ...
2
votes
2answers
36 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
41 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
0answers
19 views

Eigenvalues of a rank 2 tensor defined by an integral

I've been given the question: "Consider the tensor: $$ C_{ij}=\int_{V}{x_ix_j|\mathbf {x}|^2 + x_ix_j(\mathbf {x.n})^2} dV $$ where V is the volume of a sphere radius R centred on the origin. What ...
1
vote
1answer
30 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
29 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
61 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
90 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
83 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
31 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
68 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
67 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
80 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
45 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
47 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
54 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
201 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
50 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
137 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
57 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
145 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
73 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
87 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
42 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
185 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
238 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
86 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
102 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
229 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 ...