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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Is $ \frac{ x^T A A x }{ 1+ x^TAx} $ is upperbounded by the biggest eigenvalue of $A$?

I read somewhere that $$ \frac{ x^T A A x }{ 1+ x^TAx} $$ is bounded by the biggest eigenvalue of $A$, where $x \in \mathbb{R}^d$ and $A \in \mathbb{R}^{d \times d}$. Anyone see why this is the ...
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14 views

What is the relation between condition number of block of matrices.

I am working on a problem in which linear equations form a symmetric matrix consisting of 2 submatrices. Let that matrix is P. It looks like $$ P =\begin{bmatrix} A & B^T \\ ...
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2answers
25 views

Calculating Eigen Values / Vectors of large matrices

I want to know how to calculate the Eigenvalues / Eigenvectors of large matrices. I am fairly confident that I can calculate the Eigen properties of matrices that of size $2x2$ but I'm confused on ...
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2answers
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Eigenvalues of checkerboard matrix

I am trying to find the eigenvalues and eigenvectors of the following 4x4 "checkerboard" matrix: $$ \mathbf C = \begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 ...
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2answers
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How to solve this eigenvector eigenvalue problem?

Let $L: P_1 \to P_1$ be the linear operator defined by $L(at+b)=-bt-a$. Find, if possible, a basis for $P_1$ with respect to which $L$ is represented by a diagonal matrix. How about this?
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How to solve this eigenvalue eigenvector problem? [on hold]

Let $L:P_1\to P_1$ be the linear operator defined by $L(at+b)=bt+a$. Using the matrix representing $L$ with respect to the basis $\{1,t\}$ for $P_1$, find the eigenvalues and associated eigenvectors ...
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1answer
20 views

jordan form, similarity and eigenvalues

a complex matrix is always diagonalizable in the jordan canonical form (right? ).also, two matrices have the same jordan form if and only if they are similar. But two matrices who have the same ...
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18 views

Eigenvalues and eigenvectors of $uu'$ [duplicate]

Find the eigenvalues and eigenvectors of the matrix $ uu'$ My attempt: Let $x \neq 0$. Then, $ x$ is eigenvector implies $$uu'x = \lambda x$$ My attempt consist to multiply at left for $x'$ ...
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19 views

Root Subspaces and Linear Operators

Let $A \in M_{n}(\mathbb{C})$. Let $L_{A}$ be an operator on $M_{n\times m}(\mathbb{C})$ where $L_{A}(X) = AX$. Obtain the eigenvalues of $L_{A}$ and the root subspaces of $L_{A}$ where $A$ is an ...
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14 views

Linear Operators and Invariant Subspaces [duplicate]

Let $A$, $B$ be linear operators on a finite-dimensional vector space $V$ over the complex numbers such that $A^{2}$ = $B^{2}$ = $I$, where $I$ is the identity. Show that there exists either a ...
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1answer
32 views

Eigenvalues of nilpotent matrices

I have these two claims for a real $k\times k$ matrix $A$ 1 If $A^n=0_{k\times k}$ for some $n\in\mathbb N$ and $\lambda$ is an eigenvalue of $A$, then $\lambda = 0$. 2 If $A^n=0_{k\times k}$ ...
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26 views

Definition of Multiple .

Definition of multiple is : In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for ...
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How to prove the following about eigenvalues

Let $\mathbf{M} = [m_{ij}]$ be a symmetric matrix of size $m\times m$ of real elements. Let $\mathbf{A} = [a_{ij}^R + ia_{ij}^I]$ be a random Hermitian matrix whose elements have variance, $\sigma^2$, ...
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2answers
74 views

Conditions for a Matrix to be Diagonalizable

Let $M$ be a matrix with the entries $a_{1}, ..., a_{n}$ on the secondary diagonal (the one that ranges from $m_{n1}$ to $m_{1n}$) with all other entries being $0$. Find under which conditions the ...
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0answers
20 views

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
15 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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2answers
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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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2answers
26 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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17 views

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
27 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
48 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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16 views

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
24 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 [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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1answer
23 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
38 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
19 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 [closed]

Can anyone show an example where eigenvalues and eigenvectors are used in differential equations, more details please, thank you!
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25 views

How to solve problem in eigenvalues and eigenvectors [closed]

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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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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25 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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40 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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28 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 ...
2
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1answer
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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1answer
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 ...
2
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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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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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1answer
29 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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3answers
49 views

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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70 views

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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1answer
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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 ...