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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General structure of a 3 by 3 persymmetric matrix with zero eigenvalues

Here is an interesting problem that might be extended to higher dimensions, I am looking for different simple ways of describing a 3 by 3 matrix with one or two zero eigenvalues. This persymmetric ...
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15 views

Intuitive understanding of Maximin principle

From the the book page $324$, does someone could explain to me the Theorem $2$. Maximin principle? I have a bit of difficulties to well understand how works this theorem. A simple example would be ...
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15 views

Sum of powers of eigenvalues

The sum of the eigenvalues $\lambda_k$ of an $n\times n$ matrix is equal to the trace of the matrix, i.e. $$\sum_{k=0}^{n-1}\lambda_k=\text{tr}(A).$$ Is there a "closed form" sum of positive integer ...
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Defective eigenvalues problem - determining defect given a relation among elements, deducing third linearly independent eigenvector, etc.

Suppose I have the following matrix: $$\begin{bmatrix} 1 & 0 & 0 \\ 1 & 3 & 1 \\ -2 & -4 & -1 \\ \end{bmatrix}$$ The only eigenvalue of which ...
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2answers
21 views

Eigenvalue of a matrix and a polynomial of that matrix

Let $A$ be a $n \times n$ matrix over $F$, and let $c_1, ... c_n$ be its eigenvalues. Show that for every polynomial $g(x) \in F[x]$, the eigenvalues of $g(A)$ are $g(c_1), ... , g(c_n)$. I think by ...
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29 views

eigenvalues of $g(A)$

Let $A\in M_n(F)$ and let $c_1,\dots,c_n$ be eigenvalues of $A$. Prove that for each polynomial $g(x) \in F[x]$, eigenvalues of $g(A)$ are $g(c_1), \dots g(c_n)$. (Hint: triangulate $A$) I don't ...
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1answer
28 views

finding eigenvector from 3x3 matrix

I have $$A = \begin{bmatrix} 5 & -2 & 1 \\ -2 & 2 & -2 \\ -1 & -2 & 5 \end{bmatrix}$$ which has eigenvalues $\lambda_1 = \lambda_2 = 6$ and $\lambda_3 = 0$. I want to find ...
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If $\lambda$ is the eigen-value of a $n\times n$ non-singular orthogonal matrix $A$, then prove that $\frac{1}{\lambda}$ is also an eigen-value.

QUESTION: If $\lambda$ is the eigen-value of a $n\times n$ non-singular matrix $A$ and $A$ is a real orthogonal matrix, then prove that $\frac{1}{\lambda}$ is an eigen-value of the matrix ...
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84 views

Calculating SVD By Hand

When calculating the SVD of the matrix $$A = \begin{bmatrix}3&1&1\\-1&3&1\end{bmatrix}$$ I followed these steps $$A A^{T} = \begin{bmatrix}3&1&1\\-1&3&1\end{bmatrix} ...
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37 views

If $A$ is a matrix with negative eigenvalues, then $\exists M$ : $A = -MM^T$

Let $A$ be a symmetric matrix with all its eigenvalues negative. Prove that there exists a matrix $M$ such that : $A = -MM^T$. Now, regarding my question, I have found another older question, that ...
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38 views

Generalized eigenvector space with $\lambda_1=\lambda_2=\lambda_3=2$

Let be $$M:=\begin{pmatrix} 1 & 1 & 1 \\ -1 & 3 & 1 \\ 0 & 0 & 2 \end{pmatrix}$$ The characteristic polynomial would then be ...
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1answer
28 views

Kernel and geometric multiplicity relation

Say I have a square matrix $A$ with one eigenvalue $\lambda_1$. The minimal polynomial is $(\lambda-\lambda_1)^k$ and $\dim(\ker(A))=\alpha$. What can I know about the geometric multiplicity of ...
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1answer
35 views

Eigenvectors of nilpotent $3 \times 3$ matrix

Let $N$ be a $3\times3 $ matrix such that $N^2 = O_3$, then how many linearly independent eigenvectors will $N$ have? $N^3 = O_3$, then how many linearly independent eigenvectors will ...
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Number of Linearly Dependent Rows/Columns and Number of Zero Eigenvalues

The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns). A square matrix $A_{~ n ~ \times ~ n}$ is non-singular only if its rank is equal to ...
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1answer
44 views

Numerical Calculation of Eigenvalues of a large real Symmetric tridiagonal matrix

If I have an $N \times N$ matrix where every entry is zero except for along the super-diagonal and sub-diagonal, where the each entry is the conjugate of the last, like the following $5 \times 5$ ...
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7 views

Problem on Principal Component Analysis (P.C.A.)

Let $X \; = \; (X_1, X_2, \ldots, X_m)^T$ and $Y \; = \; (Y_1, Y_2, \ldots, Y_n)^T$. Let, $S$ = pooled variance-covariance matrix obtained from $X$ and $Y$. Let, $\alpha$ = principal component ...
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62 views

Eigenvalues and eigenvectors of the Householder matrix $H = I - \frac{2}{u^Tu} uu^T$

So during my first revision for the semester exams, I went through exercises in books/internet and I found 2-3 that caught my eye. One of them was the following: Let $u \in \mathbb R^n$ be a ...
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19 views

Same vector space for arbitrary independent vectors?

If we use n linearly independent vectors x1,x2...xn to form a vector space V and use another set of n linearly independent vectors y1,y2...yn to form a vector space S, is it necessary that V and S are ...
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3answers
62 views

What is the significance of reversing the polarity of the negative eigenvalues of a symmetric matrix?

Consider a full rank $n\times n$ symmetric matrix $A$ (coming from a set of physical measurements). I do an eigendecomposition of this matrix as $$A = E V E^T$$ Most of the eigenvalues are positive, ...
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48 views

$I_m -AB$ ivertible if and only if $I_n-BA$ invertible

Let $A$ and $B$ be $m\times n$ and $n\times m$ matrices respectively. Prove that if $\lambda$ is a non-zero eigenvalue of $AB$ then it is also an eigenvalue of $BA$ Prove that $I_m-AB$ ...
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22 views

What is the eigenvector here?

To find the eigenvector $(x_1,x_2)$, I reduce $Ax=\lambda x$ to $$\begin{align} (d_0+v_{01}-\lambda_1)x_1-v_{01}x_2=0\\ -v_{10}x_1+(v_{10}-\lambda_1)x_2=0,\\ \end{align}$$ where ...
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31 views

Self-adjoint operators and eigenvalues

In a previous exam the following question was asked which I was unable to answer due to the lack of knowledge of self-adjoint operators. Let $S$ be a self-adjoint operator on a real finite ...
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Special properties of characteristic polynomials

Is there any special property of characteristic polynomials of 0-1 (M*N size) matrix with (>=1) one row with a single non-zero entry (every column has few, say only 1-in-every-p non-zero entry)?
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characteristic polynomial problem

Suppose that $A \in \mathbb{R}^{n \times n}$ and consider the linear map $L_A : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ defined as $L_A(X) = AX$. Show that the characteristic ...
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1answer
47 views

Proof about characteristic polynomial

Suppose that $A \in \mathbb{R}^{n \times n}$ and consider the linear map $L_A : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ defined as $L_A(X) = AX$. Show that the characteristic ...
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2answers
24 views

Dimension of eigenspace of a transpose

Show that $A$ and $A^T$ have the exact same eigenvalues and that for each eigenvalue we have $\dim (N(A-\lambda I)) = \dim (N(A^T-\lambda I)) $ So this proof has basically two parts. The first ...
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1answer
31 views

Determining the eigenspace of the matricial representation of a linear transformation

I have a very generic question about determining eigenspaces because I'm a bit confused... Determining the eigenvalues of a linear transformation is the same as determining the eigenvalues of a ...
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2answers
53 views

Find all similar matrices to diagonal matrix

The given task is to find all 2x2 Matrices A that are similar to: a) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ b) $\begin{bmatrix} 1 & 0 \\ 0 & 1 ...
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Is there any way to know the algebraic multiplicity of the $0$ eigenvalue in the minimal polynomial when the rank is $1$? [duplicate]

Say I have a matrix $A$ of $r=rank(A)=1$ I know that in the characteristic polynomial the algebraic multiplicity of $(\lambda-0)$ is $n-r$ which in my case is $n-1$ Is there a rule about the ...
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Difference equations and the characteristic polynomial

The context for this is solving the gambler's ruin problem using linear algebra. I haven't found a good explanation for why the linear combination of the eigenvalues for the matrix representing a ...
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33 views

Eigenvectors of matrix equation $AX+XB^\text{T}=\lambda (CX+XD^\text{T})$

We have the matrix eigenvalue problem $$AX+XB^\text{T}=\lambda (CX+XD^\text{T})$$ Where $\lambda$ is the eigenvalue, $X$ is $m\times n$ and plays the role of the eigenvector, and $A$, $B$, $C$, and ...
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Prove that eigenvalues of the operator lie in the interval [on hold]

Let $\phi$ and $\psi$ be two self-adjoint linear operator in Euclidean space, the eigenvalues of which lie respectively in the intervals $[c; d]$ and $[m; n]$. Prove that eigenvalues of the operator ...
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29 views

Strauss's book - Weyl's asymptotic law for eigenvalues

Someone told me that in the book Partial Differential Equations by Strauss you can find a proof of Weyl's asymptotic law for eigenvalues (one can hear the volume and dimension of a domain). Is there ...
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1answer
107 views

Eigenvectors of “weighted” Hermitian matrix?

Consider two real matrices $\boldsymbol{H}$ and $\boldsymbol{D}$ with the following properties: $\boldsymbol{H}$ is a symmetric matrix (since it is a real matrix this is equivalent to being ...
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56 views

Express eigenvectors of $A^{-1}$ in terms of eigenvectors of $A$

I know the eigenvalues of the matrix $A^{-1}$ are $\frac{1}{\lambda_n}$ where $\lambda_n$ are the eigenvalues of $A$. I didn't know their eigenvectors were related; in what way are they related? Also ...
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Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
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Can a matrix satisfy all three of the following properties?

Consider an $n \times n$ matrix of the form $$ A = \begin{bmatrix} a_1 & a_2 & \ldots & a_{n-1} & a_n \\ 1 \\ & 1 \\ & & \ddots \\ & & & 1 \end{bmatrix} $$ for ...
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Eigen-values of a matrix $P^{-1}AP$

QUESTION: If A and P be $2$ non-singular $n\times n$ matrices and $\lambda$ is the eigen-value of $A$, then show that $\lambda$ is also the eigen-values of a matrix $P^{-1}AP$. I could simply ...
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3answers
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How can I prove: $\forall \lambda \in C$ with $|\lambda| = 1$, exist unitarian matrix $B$ with eigenvalue $\lambda$

How can I prove: $\forall \lambda \in C$ with $|\lambda| = 1$, exist unitarian matrix $B$ with eigenvalue $\lambda$. I tried to find a counter-example and I was not succeeded. I believe I need to ...
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1answer
25 views

Find spectral theorem of $A$ and find its singular eigenvalues.

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and ...
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Eigenvectors of Generalized Sylvester Equation $AX+XB^\text{T}=\lambda CXD^\text{T}$

Ok here's what I mean with the Sylvester equation eigenvectors. The simplest case, where $C = D = I$, has already been solved in the literature (Matrix Calculus by W.H. Steeb). $$A X + X ...
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Eigenfunctions of an integral operator

Let $Tf(x):=\int_0^x f(t)dt$ be an integral Operator ($T:L_2[0,1]\rightarrow L_2[0,1]$). I am trying to find the eigenvalues and eigenfunctions of $S:=T^*T:L_2[0,1]\rightarrow L_2[0,1]$. So far I know ...
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*Slightly* Altered Eigenvalue Problem

Consider the 2 Matricies: $$ A=\left( \begin{array}{ccc} 3 & 1 \\ 6 & -2 \end{array} \right) $$ $$ B=\left( \begin{array}{ccc} -1 & 1 \\ 0 & 0 \end{array} \right) $$ Find $x\in ...
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Eigenvalues of a fourth order differential equation with boundary conditions.

I am stuck trying to figure out the following eigenvalue equation. $x, y$ exist from $[0,T]$, and vanish at the end points. $\lambda$ is the eigenvalue. $$\dfrac{d^4x}{dt^4} + (\omega^2 ...
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1answer
18 views

Spectral Graph Theory :Cartesian product of Laplace Matrix

Let $G\times H$ be the Cartesian Product of $G$ and $H$. Determine $L(G\times H)$ in terms of $L(G)$ and $L(H)$ where $L(G) $ denotes Laplacian Matrix of $G$. Also find the eigen ...
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Bounding the off-diagonal entries of a matrix

The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $Y = \begin{bmatrix} 0 & -i \\ i & 0 ...
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First eigenvalue of laplacian

I know the laplacian $\Delta$ has only positive eigenvalues, but why there is a first one? Assume $\Delta$ is acting on an appropriate set of real valued functions on the bounded domain $\Omega ...
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How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$?

How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$? I tried with $\det(A - aI) = (\cos\phi - a)^2 + \sin^2 \phi = 0$ and I got somehow to $2\cos\phi = a$, and I believe ...
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Tricks for quickly reading off the eigenvalues of a matrix

I noticed that some mathematicians have an uncanny ability to identify the eigenvalues of matrices without doing much in the way of computation. For instance, one might notice that all the rows have ...
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1answer
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Given $v \in C^n$ that $u^Hu = 1$, and $D = iuu^H$ find all eigenvalues of $D$

Given $v \in C^n$ that $u^Hu = 1$, and $D = iuu^H$ find all eigenvalues of $D$ Well, I believe that $D$ is composed of orthonormal vectors, because of $u^Hu = 1$. Which means I believe that all ...