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 $\lambda x - Tx = 0$.

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linear homogeneous constant coefficient systems

Solve the following LHCC system by finding the eigenvalues, eigenvectors and generalised eigenvectors. Give a fundamental set of solutions and show that the set is independent. $$x'= \left[ ...
4
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1answer
26 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as ...
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0answers
42 views

eigenvalues of $A^TA$ and $AA$

I am a little bit confused about such fundamental problems: Suppose 1. $Ax=\lambda x$. 2. $A \in \mathbb{R^{n \times n}} $. Case I: $$A^TAx = \lambda A^Tx=\lambda \lambda x=\lambda^2x$$ ...
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Linear Algebra: Guidance on a Eigenvalue/Eigenbasis problem, please?

Here's the problem, but I only need some help with part C: http://i.imgur.com/UwRBGIO.png This is the information and answers from the back of the book: http://i.imgur.com/BFs2z2s.png I understand ...
2
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2answers
37 views

Eigenvalues of composition of functions

I am trying to do the following exercise: Let $V$ be a $K$-finite dimensional vector space and let $f,g \in Hom(V,V)$. Define $Spec(f)=\{\alpha \in K / \alpha \space \text{is an eigenvalue of f}\}$. ...
2
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1answer
30 views

Building matrices from eigenvalues

I saw a question some time ago, asking about the eigenvalues of the matrix $$A=\begin{pmatrix}5&-3&0\\-3&5&0\\0&0&2\end{pmatrix}$$ which were then shown to be ...
1
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1answer
15 views

Kernel Principal Component Analysis (PCA)

I learn kernel PCA from wikipedia. In this article, the eigen equation is \begin{equation} N \lambda \vec{\alpha} = \boldsymbol{K} \vec{\alpha} \end{equation} where $\lambda$ is the eigen value, ...
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0answers
13 views

How to judge eigenvalues from leading diagonal? [on hold]

If we have a 3 x 3 matrix with 3 pivot columns. How can we say that the leading diagonals coefficients are the eigenvalues of the matrix? Can someone please explain..Can't visualise this thing.
3
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0answers
43 views

Find the number of distinct real values of $c$ such that $A^2x=cAx$

Let $$A= \begin{pmatrix} 5 & -3 & 0 \\ -3 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$ and $c$ be a real no. such that $A^2x=cAx$ for some non-zero vector $x$. Then the number of ...
1
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1answer
19 views

matrix transformation - eigenvector

I am trying to understand eigenvectors. An Eigenvector is nothing more than a vector that points to some place. This pointing vector will then be invariant under linear transformations. Now my ...
0
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1answer
27 views

finding angle and scalar c

matrix $$A = \pmatrix{ 4&-5\\5&4}$$ is standard matrix of a linear transformation from $R^2 \to R^2$ that consists of a rotation through an angle composed with multiplication by a scalar $c.$ ...
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14 views

Given the matrix representation what is the expectation value

For a particle with spin $\frac{3}{2}$, construct the matrix representation for $S_z, S_x$ and $S_y$. If the particle is in an eigenstate of $S_z$, what is $\langle S_x\rangle$ and $\langle ...
1
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1answer
29 views

Etingof problem 2.15.1 Representations of sl(2)

I'm studying from Etingof's Introduction to Representation Theory. This is problem 2.15.1, part a. I feel I'm close to the solution. Here's what I have. Problem: A representation of sl(2) is a vector ...
2
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3answers
59 views

Find an arbitrary power of a lower triangular matrix of size $3\times 3$

Let $F$ be a field and let $A=\begin{bmatrix}a&0&0\\1&a&0\\0&1&a\end{bmatrix}\in\mathscr{M}_{3\times 3}(F)$. Show that ...
0
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0answers
10 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
3
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1answer
26 views

How to prove this identity involving characteristic polynomials on both sides?

Suppose $A\in \Bbb C^{m\times n},B\in \Bbb C^{n\times m},m\ge n$, prove: $$\det(\lambda I_m-AB)=\lambda^{m-n}\det(\lambda I_n-BA)$$ I don't want to get into nasty determinant calculation. Instead, I ...
0
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0answers
17 views

eigenvalues as an optimization problem

I am thinking on how to compute eigenvalues as the solution of an optimizing problem. Until now I can think of an optimizing(minimizing) problem as following. As we know, eigenvalues of a matrix ...
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0answers
27 views

Why does $\|A\|^2_2 \geqslant \|Av^k\|^2_2$ where $\lambda_k$ is the largest eigenvalue of $A^TA$ [on hold]

Could someone explain why: Let $\lambda_k$ be the largest eigenvalue of $A^TA$, then $$\|A\|^2_2 \geqslant \|Av^k\|^2_2$$ ($v^k$ is the eigenvector corresponding to $\lambda_k$) From a ...
0
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2answers
32 views

what do eigenvalue & eigenvector of $4\times4$ matrix represent?

What do we get when calculating the eigenvalue and eigenvector of a $4\times4$ matrix? What do those values actually represent?
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2answers
22 views

Let $A$ be a complex $2$ by $2$ matrix having distinct eigenvalues $a, b$. Show that $A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A - aI)$.

Let $A\in\mathscr{M}_{2\times 2}(\mathbb{C})$ be a matrix having distinct eigenvalues $a\neq b$. Show that, for all $n > 0$, \begin{equation*} A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A ...
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1answer
24 views

A basic question about eigenvalue

Suppose a symmetric matrix $A$ is of dimension $N \times N$. Then the largest eigenvalue of $A$ is equal to $\max_{i} \sum^{N}_{j=1} |A_{ij}|$. Is this statement true? If so, how shall I show it ...
3
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1answer
41 views

Characterize magic matrices in terms of their eigenvalues. A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$.

A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$. Characterize magic matrices in terms of their eigenvalues. I know that $c$ is an egenvalue and ...
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1answer
18 views

Let $D$ be a nonsingular diagonal matrix. Show that $1\notin spec(DA)$ if and only if $D - A$ is nonsingular.

Let $F = \mathbb{F}_3$ and let $n$ be a positive integer. Let $D = [d_{ij} ]\in\mathscr{M}_{n×n}(F)$ be a nonsingular diagonal matrix and let $A\in\mathscr{M}_{n×n}(F)$. Show that ...
0
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1answer
36 views

Questions about Eigenspace

I'm learning about Eigenspaces and have a few questions. Do eigenspaces, eigenvalues, and eigenvectors correspond to a tranformation or can a single vector space $V$ have an eigen-stuff? Is an ...
3
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2answers
60 views

How to determine generalized eigenvectors of $\begin {bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 &1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$

I want to calculate the general solution of this DE-system: $$ \frac{d \vec x}{d t}= A \vec x,\text{ with }A = \begin {bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 ...
1
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1answer
25 views

prove that matrix is diagonal by matrix rank and eigenvalue rank

$A$ is matrix $9\times9$ with rank of $5$, there is rank$(A-3I)=5$, the matrix has another eigenvalue of 5. I need to prove that $A$ is diagonal and find the similar diagonal matrix of $A$. I'm stuck, ...
0
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3answers
33 views

Find the complex eigenvectors, knowing the eigenvalues

If $$A= \begin{pmatrix}1 & -1 \\ h^2 & 1\end{pmatrix},$$ I know the complex eigenvalues are $1+ih$ and $1-ih$. How do we find the complex eigenvectors? Can someone please explicitly show me ...
0
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1answer
35 views

Properties of a matrix that shares the set of real eigenvalues with its inverse

For a $3\times 3$ real matrix, let $c(A)$ denotes the set of real eigenvalues of $A$. Suppose $c(B)=c(B^{-1})$ for a non-singular matrix $B$ with no repeated eigenvalues. Then which of the following ...
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0answers
31 views

Find the matrix $P$

$A= \begin{bmatrix}1 & -2 & 3\\-2 & 6 &-9 \\3 & -9 & 4 \end{bmatrix}$ Find $P$ with non-negative integer entries and has determinant $2$. $P^TAP=\begin{bmatrix}a & 0 ...
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3answers
179 views

If $\,A^3-A+I=0,\,$ then $A$ is invertible

Prove or disprove. If $A$ is a square matrix and $A^3-A+I=0,$ then $A$ is invertible. Is it possible to say the characteristic polynomial of $A$ is $\,p(t)=t^3-t+1$, and $A$ is invertible since ...
0
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1answer
18 views

Effect of spectral shift on the eigenvalues of a real symmetric matrix [duplicate]

Suppose a matrix A(real symmetric) is changed to A − σ I, where σ is any scalar quantity and I is the identity matrix. Explain what happens to the eigenvalues and eigenvectors of A? I am unable to ...
0
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0answers
18 views

Finding the Eigenvalue of a general transformation

I'm studying for my Linear Algebra final and I'm having some issues with this proof: Given $T:V\rightarrow V$ $T=T^2$ what are all the possible Eigenvalues for T I'd appreciate any help you care ...
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2answers
38 views

eigenvalues of A - aI in terms of eigenvalues of A

I am stuck with this question of my assignment where given that A is nxn square matrix and a be a scalar it is asked to - Find the eigenvalues of A - aI in terms of eigenvalues of A. A and A - aI ...
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2answers
50 views

What are the possible eigenvalues of matrix $A$ that satisfies $A^2=-I$? [closed]

Let $A$ be a matrix such that $A^2=-I$, where $I$ is identity matrix. What are the possible eigenvalues of $A$?
3
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1answer
28 views

When an eigen vector is zero vector

Question : Mike opens a bank account with an initial balance of 2000 dollars. Let b(t) be the balance in the account at time t. Thus b(0)=2000. The bank is paying interest at a continuous rate of 3% ...
2
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4answers
149 views

Minimal polynomial for an invertible matrix and its determinant

So here's one that I can't quite crack: Let $A\in M_n(\mathbb{F})$ be an invertible matrix with integer eigenvalues. Its minimal polynomial is ...
1
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1answer
20 views

No minimal polynomial for differentiation operator

Let $D$ be the Differentation operator of the of polynomials over $R.$ Prove that there is no polynomial $g(t),$ such that $g(D)=T_0.$ But characteristic polynomials satisfies it's operator. I dnt ...
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3answers
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Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$.

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$. here is the source of the problem. In the last ...
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3answers
34 views

Why is this matrix invertible [duplicate]

I was wondering if there is a way to see why $(1+A)$ invertible, if $A$ is a skew symmetric matrix. and I know that all eigenvalues of $A$ have zero real part and $A$ is unitarily diagonalisable.
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3answers
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What are the possible eigenvalues of a linear transformation $T$ satifying $T = T^2$ [duplicate]

Let $T$ be a linear transformation $T$ such that $T\colon V \to V$. Also, let $T = T^2$. What are the possible eigenvalues of $T$? I am not sure if the answer is only $1$, or $0$ and $1$. It holds ...
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When do a Regular graph has an odd eigenvalue?

Merely looking at adjacency matrix of a regular graph, without explicit calculation, can we decide that graph will have an odd eigenvalue or not? If regularity is odd, we are sure that it will be an ...
1
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1answer
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Find Matrix $S$ such that $S'CS=\left(\begin{array}{cc} 1_{N-1} & 0 \\ 0 & 0 \end{array} \right)$ where $C:=1_N-\iota \iota'$

The Centering Matrix $C:=1_N-\iota \iota'$ has eigenvalue $1$ of multiplicity $n − 1$ and eigenvalue $0$ of multiplicity $1$. Therefore a matrix $S$ with columns consisting of eigenvectors of $C$ can ...
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24 views

Eigenvectors of a real positive semi-definite submatrix

Let $\mathbf{A}$ be a real positive semi-definite matrix, and $\mathbf{V}$ and $\mathbf{\lambda}$ its eigenvectors and eigenvalues, respectively. I am wondering what is the relationship between ...
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23 views

Inital Value Problem from general solution

We have the following matrix: $\frac{d\mathbf{Y}}{dt} = \begin{pmatrix} -3 & -5 \\ 3 & 1 \end{pmatrix}\mathbf{Y}$ And the inital condition: $\mathbf{Y_0} = (4,0)$ I have got the correct ...
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0answers
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How to force a system of linear equations to return a non-trivial solution instead of the trivial one when finding the eigenvectors

When you've found the eigenvalues for a matrix A. And insert each eigenvalue into the relation: $(A-\lambda I) = 0$ and row reducing to echelon form for each one. In the situations where you end up ...
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2answers
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characteristic polynomial and eigenvalues [closed]

Find the characteristic polynomial and eigenvalues of $A=\left[\begin{matrix} 3 & 1 & 1 \\ 0 & 5 & 0 \\ -2 & 0 & 7\end{matrix}\right]$ $$ \begin{align*} \det(A-\lambda I) ...
3
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1answer
49 views

Easy Problem:Find eigenvalues of a $3 \times 3$matrix

Let $ A = \left[ {\begin{array}{cc} 1 & 1 & 1 \\ 1 & w^2& w \\ 1 & w & w^2 \end{array} } \right] $ where $w$(other than 1) is a cube root of unity.Let ...
4
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2answers
28 views

Find the eigenvectors corresponding to an eigenvalue

I know how to find the eigenvectors corresponding to an eigenvalue of a matrix $A$: we basically need to find the vectors of the nullspace of $\lambda I - A$, but in my case, I have a matrix $A$ like ...
1
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2answers
29 views

When do two linear hermitian operators have a common eigen vector?

Let $H$ be an finite dimension hilbert space. Let $L_1$ and $L_2$ be two hermitian linear operators acting on this space. I know if these two operators commute they can be diagonalized in a common ...
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2answers
43 views

Find the eigenvalues for a matrix which is a product of matrices

Suppose I have a matrix $A \in \mathbb{R}^{2, 2}$ which is the product of $3$ other matrices, lets call them $A_1 = \left(\begin{matrix} cosx & -sinx \\ sinx & cos x\end{matrix}\right)$, $A_2= ...