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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Proving the eigenvalues of a real symmetric matrix are real [duplicate]

Can I prove that, if $A$ is a real symmetric matrix, that its eigenvalues are real? Can I also prove that the eigenvectors associated with distinct eigenvalues of $A$ are orthogonal? I'm not sure ...
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1answer
13 views

Regarding the maximum eigen value

In a paper, the author removed the matrix $P$ and use the maximum eigenvalue multiplied by identity matrix , so is the following true? $$x^T P x \le x^T \bar\lambda(P) I x$$ where $x\in\mathbb ...
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relation eigenvalue and adj(A-λI)

Let $A$ be a matrix in $\mathbb C^{n×n}$, let $λ$ be an eigenvalue of $A$ with eigenvector $x$. Why is there some $y \in \mathbb C^n$ such that $adj(A−λI)=x{y^*}$?
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In two dimension space, why two eigenvector cannot be at right angels to each other?

When I read the wikipedia about eigenvector, it said: If two-dimensional space is visualized as a rubber sheet, a linear map with two eigenvectors would be a stretching along two directions ...
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25 views

Proving A ~ CB if A = BC and 0 is not an eigenvalue of B

I am trying to prove that, if A, B, and C are $n \times n$ matrices, $0$ is not an eigenvalue of $B$, and $A = BC$ that $A$ is similar to $CB$ I know that I have to get to $A = SCBS^{-1}$ for some $n ...
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finding formula for M^n

Find formulas for the entries of M^n, where n is a positive integer. M = [10 10] [-5 -5] My try: I found eigenvalues and eigenvectors, and put A as set of two eigenvectors and D as ...
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1answer
24 views

Matrix finding $A=SDS^{-1}$

$\displaystyle A= \begin{bmatrix}-18 & 10 \\ -20 & 12 \end{bmatrix}$ Find $S$, $D$, $S^{-1}$ such that $A = SDS^{-1}$ I used eigenvalues for $D$ and eigenvectors for $S$ but not getting ...
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2answers
42 views

n x n matrix has the same eigenvalues as its transpose

I am trying to prove that a $n \times n$ matrix $A$ and $A^T$ have the same eigenvalues. I can prove that $A$ and $A^T$ have the same entries on the diagonal, but I am not sure where to go from ...
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1answer
28 views

Proving that eigenvalues are real and eigenvectors are orthogonal

If $A$ is a real symmetric matrix, can I prove that all of the eigenvalues of $A$ are real and that all eigenvectors associated with distinct eigenvalues are orthogonal? If so, where do I start to ...
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Help understanding Wiener filtering formula

I would like some help interpreting the following formula, equation 1 from this paper: https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/strela.pdf $\hat{X} = ...
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1answer
22 views

Proof of positive semidefinite projection [on hold]

How to show the sol. of $\min \limits_{X \in \mathbb{S}^+}||X-C||_F^2$ is $U \hat \Lambda U^T$ where $\hat \Lambda = diag(max(0,\lambda_1), ... , max(0,\lambda_N))$, $C = U\Lambda U^T$ and $\Lambda ...
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Spectral radius of matrix from SOR method

Suppose we write a matrix $A = L + D + U$ with lower triangular, diagonal and upper triangular parts. When trying to solve the equation $Ax=b$, we use a successive overrelaxation technique such that ...
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solve problem in adjoint - eigenvalue

let $A$ is a matrix $n \times n$ and let $\lambda$ is eigenvalue and $x$ is eigenvector. why there is some $y \in \Bbb{C}^n$ such that ${\rm{adj(A - }}\lambda {\rm{I) = x}}{{\rm{y}}^*}$ «Please help ...
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1answer
18 views

show invariant subspace is direct sum decomposition

Let $f \in End(V)$ ($V$ is a finite dim.) be diagonalized where $a_1, … ,a_k$ are eigenvalues and for $i \neq j$ we have $a_i \neq a_j$. Prove for every subspace $f$ invariant $W \subset V$ holds ...
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1answer
37 views

For real matrices, if $A$ and $B$ are both positive-definite, show that all of $AB$'s eigenvalues are positive.

The original question goes equivalently like this For real matrices, if $A$ and $B$ are both positive-definite Prove: all the eigenvalues of $AB$ are positive. Facts that I know may have a ...
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1answer
14 views

Efficiently compute the eigenvectors of the Laplacian of a symmetric positive matrix

I am working with a matrix A relatively large (200k x 200k), and I want to compute the eigenvectors of the Laplacian: $L = D - A^2$, where $A$ is symmetric. I don't need all eigenvectors, just a few ...
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Where eigenvalues and eigenvectors are used in differential equations [on hold]

Can anyone show an example where eigenvalues and eigenvectors are used in differential equations, more details please, thank you!
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How to solve problem in eigenvalues and eigenvectors [on hold]

let A is a matrix $n \times n$ and let $\lambda$ is eigenvalue and X is eigenvector why there is some $y \in \Bbb{C}^n$ such that ${\rm{adj(A - }}\lambda {\rm{I) = x}}{{\rm{y}}^*}$ please help ...
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10 views

How to find Eigenvectors of a non symmetric matrix using QR decomposition?

The QR method efficiently calculates the eigenvalues of a matrix. If I try to find eigenvectors alongside the eigenvalues($Q_1Q_2.....Q_n$), the results seem to be correct only for symmetric matrices. ...
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1answer
17 views

Name of specific symmetric Toeplitz matrix

Is there a name for a Toeplitz matrix, which has all diagonal elements equal to let's say a and all off-diagonal elements equal to let's say b? Also, is there any general proof for the eigenvalues of ...
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24 views

Is my proof about minimal polynomial correct

I am to prove that the characteristic polynomial and minimal polynomial have same roots. That is, if $\lambda$ is an eigenvalue of the linear transformation $T$ and if $p(t)$ is the minimal polynomial ...
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38 views

What if generalised eigenvector is the zero vector

I have a 3*3 matrix A= $$\begin{pmatrix} 3 & 0 & -1 \\ -1 & 2 & 1 \\ 1 & 2 & 3 \\ ...
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25 views

An inverse of Jordan matrix - basis

Let $A\in M_{n\times n}$ be and invertible matrix over complex field and we assume it's already at Jordan form where $B=\{v_1,…,v_n \}$ is Jordan basis for A. Find Jordan form and Jordan basis for ...
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61 views

Why is $0$ not an eigenvalue?

I have the following Sturm-Liouville problem \begin{cases} u''+\lambda u =0 & 0 < x < 1, \\ u(0)-u'(0)=0, & u(1)+u'(1)=0, \end{cases} and I am trying to show that all the ...
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19 views

Prove that the only nilpotent operator is 0? [duplicate]

I need to prove that if $\phi:V \to V$ is nilpotent, then its only eigenvalue is $0$. I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How would I ...
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1answer
21 views

Spectrum and Bases of Eigenspaces [Resolved]

$f: F_3^2 \to F_3^2$, given by $f(a, b) = (b, -a-b)$ Spectrum: roots of det(f-xI) A = [f(1,0), f(0,1)] A=[(0,-1), (1,-1)]
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What would be a characterization of a definite operator?

Let $V$ be an $n$-dimensional inner product space and let's call $T\in \mathcal L (V)$ definite if $$\forall x \neq0: \langle Tx,x\rangle \neq 0. $$ An obvious sufficient condition for $T$ to be ...
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How do find the general solution to this system of differential equations?

$$ \begin{align} \frac{dI_n}{dt} &= 2(1-p) I_n - I_n + 2(1-p) I_v \\ \frac{dI_v}{dt} &= 2p I_v - 3 I_v + 2p I_n \end{align} $$ I tried to find the eigenvalues and the eigenvectors for this ...
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33 views

Eigen values of a matrix

I have a quick question about eigenvalues. If I'm given one eigenvector of a 3 by 3 matrix, I can easily calculate its corresponding eigenvalue. I know I can determine the other two eigenvalues by ...
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28 views

Finding the eigenvalues of a matrix problem

So I do know how to compute the eigenvalues of a matrix. At least, that's what I thought. I got the matrix A = \begin{bmatrix}1&-2&0\\-2&0&2\\0&2&-1\end{bmatrix} My approach ...
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Angle of rotation from complex eigenvalues of Rotation matrix

The complex eigenvalues of a Rotation matrix are $e^{-i\theta}$ and $e^{i\theta}$. Corresponding to these we get complex eigenvectors. We know that the eigenvector corresponding to the eigenvalue 1 ...
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1answer
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Quick question about relation between Nullspace and Eigenspace [duplicate]

I have a question about a note given in a linear algebra textbook. It is just given as a remark, with no proof or explanation so I want to make sure I understand it correctly. First, it gives the ...
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Matrix diagonalization - eigenvalues on diagonal

Diagonalization of a square matrix $A$ consists in finding matrices $P$ and $\Delta$ such that $A=PD P^{-1}$ where $D$ is a diagonal matrix. What theorem tells us that $P$ is a matrix composed of the ...
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Find the spectrum and bases for corresponding eigenspaces. Determine if each operator is diagonalizable.

I have this question to do on an assignment, really stuck! Any help is appreciated, thanks! For each of the following, find the spectrum and bases for the corresponding eigenspaces. Determine if ...
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How the nullspace of an eigenproblem changes

Given the eigenproblem $CAv = \lambda CBv$ a) When does dropping $C$ not change/increase the nullspace? (apart from $C$ being square and invertible) b) When does premultiplying by $D$, i.e. $DCAv = ...
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Problem from I.N.Herstein (Linear transformation)

Let $A=(a_{ij})$ be such that for each i, $\sum_{j} a_{ij}=1$. prove that $1$ is characteristic root of A. Generalization: Let $A=(a_{ij})$ be such that for each i, $\sum_{j} a_{ij}=n$ where n is any ...
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Why does this work? Fourier coefs. of function with min energy in window is eigenvector of window coef. matrix.

Let me begin with saying I have never got a good handle on eigenvectors and eigenvalues. My best hunch is that the eigenvectors are the 'best' basis for a linear transform along which the transform ...
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What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?

Given a set of points $x_1,x_2,...,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m$ x $m$ Euclidean Distance Matrix $D$ where $D_{ij}={||x_i-x_j||}^2$. We know a little bit about these ...
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Consider a linear operator $L$ and some polynomial of it, $L'=p(L)$. Show that the minimal polynomial of $L'$ has smaller degree than that of $L$.

Consider a linear operator $L$ and some polynomial of it, $L'=p(L)$. Show that the minimal polynomial of $L'$ has degree less than or equal to the minimal polynomial of $L$. First, start working over ...
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Eigenvalue dependent operator

Consider the wave equation in a Riemann metric $g^{\mu\nu}$ with spacetime off-diagonal components $g^{i0}$: ...
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Prove that the Eigenvalues of this Matrix are in [0,1 ]

Let $E,F \subset \mathbb{R^n}$ Note that $< . >$ defines the Inner product on $\mathbb{R^n}$ Let $(e_1,....,e_k)$ and $(f_1,.....f_l$) be Orthonormal bases of E and F respectively. Consider ...
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Bases of eigenvectors and diagonal representation

I just wanted to make sure I understand what's going on in this article: Source My interpretation is: suppose we have a linear transformation represented by a matrix $A$ that transforms vectors ...
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Relationship between eigenvalues of two related, Euclidean distance matrices

If $X=\{x_1,\ldots,x_N\}$ is a set of points in $\mathbb{R}^n$ then one can generate a Euclidean distance matrix $D = [d_{ij}]$ where $d_{ij}=\Vert x_i-x_j\Vert_2^2$ is the square of the Euclidean ...
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Matrix Compression

I have a large matrix made up of only 1's and 0's. I want to compress this matrix into a smaller matrix for storage but need to be able to reproduce the original matrix. My thought is to use ...
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Change of coordinates of a matrix $A$ to a basis formed by its eigenvectors

What does it mean to change coordinates of a matrix? A matrix is just a bunch of numbers. All we can do is try to experss it as a linear combination of some other matrices of the same size. I don't ...
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zeros of $x^*Ax$, a quadratic form

The question hopefully says it all! We have a Hermitian matrix $A=A^* \in \mathbb{C}^n$ and a quadratic form: $f(x)=x^*Ax,~x\in \mathbb{C}^n$ We want to find the solution of $f(x) = x^*Ax = 0$ When ...
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Help me with this.

Let $a=2i -j+ k$, $b =i +2i- k$ and $c= i + j -2k$ be three vectors. If the vector $d$ is on the plane of $b$ and $c$ and its projection on $a$ is of magnitude $\sqrt{2/3}$, find $d$ later: So I know ...
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distribution of eigenvectors of a random matrix

Let $\mathbf{A}\in\mathbb{C}^{n\times n}$ be a random hermitian matrix. Assume that the eigenvalues of this matrix have continuous probability distribution. 1.Can we say that the eigenvectors ...
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Does an upper triangular matrix always have $\langle 1,0,0\rangle$ as one of its normalized eigenvectors?

My question is exactly what the title asks. An example matrix to mess around with is: $$\left\{ \begin{matrix} 3 & 2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 5 \end{matrix} \right\}$$
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Given an eigen values evaluate $S*\tiny\begin{bmatrix} 0\\1\\0 \end{bmatrix}$

Eigen value is given as $\lambda = 2,-3,5$.. $v_{1} = \begin{bmatrix} 1\\-3\\-2 \end{bmatrix}$ $v_{2} =\begin{bmatrix} -2\\7\\5 \end{bmatrix}$ $v_{2} =\begin{bmatrix} 0\\0\\1 \end{bmatrix}$ A) ...