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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Finding a linear transformation such that $T^{3} = T $

I have to show that there exists a linear transformation such that $T^{3} = T $ i can see that from here that T has eigen values $0.1.-1$ .But how do i find linear transformation .Also for v and q ...
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3answers
41 views

Relationship between Eigenvalues

I am looking at a matrix $$\mathbf{M} = \left(\mathbf{I}+k\theta\mathbf{B}^{-1}\mathbf{A}\right)^{-1}\left(\mathbf{I}-k(1-\theta)\mathbf{B}^{-1}\mathbf{A}\right) $$ where $\mathbf{I}$ is the identity ...
2
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1answer
21 views

Eigenspaces and jordan normal form

I have a question here regarding the jordan normal form of two matrices where the eigenspace is one is contained in the other. Let $A,B$ be two $nxn$ matrices s.t $AB=BA$. I firstly proved that the ...
2
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2answers
33 views

Dimension of $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$?

What is the dimension of an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$? Do I have to make distinct cases with as $p + q < n$ and equal to $n$? And if their ...
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3answers
43 views

Determine a matrix knowing its eigenvalues and eigenvectors

I read through similar questions, but I couldn't find an answer to this: How do you determine the symmetric matrix A if you know: $\lambda_1 = 1, \ eigenvector_1 = \pmatrix{1& 0&-1}^T;$ ...
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9 views

How to Simplify/Rewrite this Expression into a Generalized Eigenvalue Problem - via Similarity perhaps?

I have the following optimization problem: \begin{eqnarray} min~b' y' Z (Z' \Omega Z)^{-1} Z' y b \end{eqnarray} such that $b'b=1$. The matrices are $Z \in R^{n \times k}$, matrices $y \in R^{n \times ...
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1answer
25 views

Why is a linear autonomous system asymptotically stable iff for all eigenvalues $\lambda$ of $A$, $Re(\lambda) < 0$

I'm trying to understand asymptotic stability of linear antonymous systems. I'm not sure if for the system $x' = Ax$, $x(t) = 0$ is the only fixed point that can be stable. In any case, I can ...
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1answer
22 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
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1answer
31 views

How to determine the signs of the eigenvalues of a symmetric $3\times 3$ matrix?

This is a homework problem: Let $a,b,c$ be positive real numbers such that $b^2+c^2<a<1$. If $A=\begin{pmatrix} 1&b&c\\b&a&0\\c&0&1\end{pmatrix}$, then which of the ...
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9 views

Algebraic multiplicity of an eigenvalue for abstract operators

How does one define algebraic multiplicity of an eigenvalue for an abstract operator? (for a matrix the definition is clear). E.g. Consider $\partial_x^2$ on $H^2_{per}(0,1)$ then $\partial_x^2 ...
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1answer
24 views

Courant minimax principle on block matrix

in going through some books about numerical mathematics I found the following exercise: Let $A,B \in \mathbb{R}^{n\times n}$ with $A$ symmetrical and rank($A$) = rank(B) = n. Define $M = ...
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1answer
39 views

Eigenvalues and eigenfunctions of fourth order ODE

Find the eigenvalues and eigenfunctions of the problem $$y^{(4)} − λy = 0$$ with the boundary conditions (i) $\quad y(0) = y'' (0) = y(β) = y'' (β) = 0$ (ii) $\quad y(0) = y' (0) = y'' (β) = y''' ...
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2answers
33 views

Proof with orthogonal matrix

I stuck at this problem: I need to prove that for linear transformation $$T:R^n\to R^n$$ defined by $$T(x)=Px$$ such that $$P^T=T^{-1}$$ for any $x,y$ $$T(x) \cdot T(y)=x\cdot y$$ and also that $T$ ...
3
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0answers
23 views

Eigenvalues of Overlapping block diagonal matrices

I look for eigenvalues of general overlapping block diagonal matrices. e.g. $$\left[ \begin{matrix} 1 & 4 & 0 & 0 & 0 & 0\\ 4 & 2 & 3 & 2 & 0 & 0\\ 0 & 3 ...
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2answers
26 views

Commuting operators

Let's consider a number of linear operators, defined on a finite dimensional complex vector space, which two by two commutes with each other. (the amount of them can be infinite). How to prove that ...
0
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1answer
24 views

One eigenvalue and eigensystem

Matrix $A \in \mathbb{K}^{n,n}$ has one engenvalue $\lambda \in \mathbb{K}$ and its engensystem $V_{\lambda}$ has dimension that equals to $n$. How to show that $A = \lambda I_{n}$?
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1answer
29 views

If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($ \lnot \#Spec= 0$)?
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2answers
23 views

Approximate the second largest eigenvalue (and corresponding eigenvector) given the largest

Given a real-valued matrix $A$, one can obtain its largest eigenvalue $\lambda_1$ plus the corresponding eigenvector $v_1$ by choosing a random vector $r$ and repeatedly multiplying it by $A$ (and ...
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0answers
23 views

Spectrum of convolution operator

I was trying to find the spectrum of the convolution operator $$ J \ast u = \int_D J(x-y) u(y) dy $$ for bounded domain $D \subset \mathbb{R}$. Does anybody know it or have a reference for me? ...
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1answer
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How do I show that $\inf\limits_{\det(X)\neq0}\|X^{-1}AX\|^{2}_{F}=\sum\limits_{\lambda\in{\Lambda}}|\lambda|^{2}$?

Show that $$\inf\limits_{\det(X)\neq 0}\|X^{-1}AX\|^2_F=\sum_{\lambda\in\Lambda}|\lambda|^{2}$$ holds, where $\Lambda(A)$ is the set containing all eigenvalues of A, and $\|\cdot\|_{F}$ is the ...
3
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31 views

(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal ...
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25 views

Using eigenvalues of a hessian matrix vs D operation to classify critical points.

Having recently covered using the discriminant, $D(x_0,y_0)$, for classifying critical points of equations of two variables. For example: $$R(x,y)=-x^2+4x+2xy+8y-2y^2$$ to find that $(6,8)$ is the ...
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1answer
27 views

Show that for every three linear maps $A, B, C: V → V$ we have $rk(ABC) ≤ rk(B)$.

Let $V$ be a vector space. Show that for every three linear maps $$A, B, C: V → V$$ we have $$rk(ABC) ≤ rk(B)$$ My only idea is to try and show something like $rk(ABC) ≤ rk(BC) ≤ rk(B)$, but ...
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19 views

finding proections with certain eigenvalues

Find a projection with following eigenvalues: $\sigma$($\emptyset$) $\sigma$(1) $\sigma$(0) I know that a shift matrix has no eigenvalues and that a zero matrix has the eigenvalue 0, but are these ...
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Show A is not similar to a Diagonal Matrix

Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $A = \begin{bmatrix} 4 & 0 & 0 &0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & -2 & -3 \\ 0 & -1 ...
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1answer
19 views

Finding eigenvalues for a vectorspace such that the matrixrepresentation is a diagonal matrix

Problem: Let $T$ be a linear operator on the vectorspace $V = M_{2 \times 2}(\mathbb{R})$ and let $T\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & b \\ c & a ...
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2answers
19 views

Convergence rate of the power method for finding eigenvectors

Let $M$ be a real-valued square matrix with an eigenvector $w$ strictly larger (in absolute value of the corresponding eigenvalue $\lambda$) than all others, and let $v$ be any vector not orthogonal ...
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39 views

eigenvalues of an involution

Let $V \neq \{0\}$ be a K-vector space and let $P : V \rightarrow V $be linear. Furthermore, let P be an involution, i.e. $P (P(x)) = x $ for every $x \in V.$ Show that if $P \neq ±id,$ then $V = ...
2
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1answer
33 views

Find eigenvalue of matrix given eigenvector

I have the following matrix: $P= \begin{bmatrix} 6 & -3 \\ 2 & 1 \end{bmatrix} $ And its eigenvector is : $v= \begin{bmatrix} 4 \\ 3 \end{bmatrix} $ I would like to find its ...
3
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1answer
36 views

Calculate a matrix to the power of “n” given an eigenvector

I have a question that I simply cannot solve. I do not want a direct answer to the question but simply an explanation as to the steps one would take to go about solving it, that way I can try it ...
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1answer
18 views

What's the relationship between the rank and eigenvalues of symmetric positive semidefinite matrix (real domain)?

Could anyone tell me the relationship between the rank and eigenvalues of symmetric positive semidefinite matrix (real domain)? According to simple algebra theorem: ...
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Find a 3x3 matrix A such that 5 is its only eigenvalue; A is non-diagonlizable; and standard vectors e1,e2 are eigenvectors of A of eigenvalue 5.

Could someone point me in the right direction? I can easily go from a square matrix to finding its eigenvalues and eigenvectors but I have no experience going the other way around.
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Symbolic expression of eigenvalues for this symmetry 3x3 matrix

Can anyone suggest if the analytical expressions of the eigenvalues for this symmetry real matrix $L$ exist or not? All variables are real. $$\begin{align} g_{11}&= ...
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2answers
22 views

Compact subset of space of matrices and compactness verification of a set of eigenvalues

Let $M_n(\mathbb R)$ be the vector space of real matrices of size $n$ , identified with $\mathbb R^{n^2}$ ; let $X \subseteq M_n( \mathbb R)$ be a compact set ; let $S \subseteq \mathbb C$ be the set ...
0
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1answer
18 views

Covariance matrix of linear transformation of eigenvector matrix

Let $K_t=E'R_t$ where $E$ is a matrix with the eigenvectors of the covariance matrix of $R_t$. According to my book, then the following holds: ...
2
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2answers
57 views

Singular matrix geometric sum

What is a fast way to calculate the sum $M + M^2+M^3+M^4+\cdots+M^n$, where $M$ is an $n \times n$ matrix whose cells are either $0$ or $1$? I have researched an alternative way which makes use of ...
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1answer
45 views

Finding the eigenvector of a matrix $2\times 2$ in two ways?

I need some help. I dont understand why the eigenvector $v_1$ of the matrix \begin{pmatrix} 0 & 1 \\ -1 & -2 \end{pmatrix} is the vector $(1,-1)$ and no the vector $(-1,1)$. $\lambda_1 = ...
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1answer
36 views

Proof that there exists a non-negative eigenvector corresponding to eigenvalue 1 of stochastic matrix

Let $P \in [0,1]^{n \times n}$ be a [irreducible or reducible] stochastic matrix where its rows sum to 1 i.e. $$ \forall i \in \{ 1 , \dots n \} \quad \sum_{j=1}^{n} P_{ij} = 1 $$ It is easy to show ...
3
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2answers
84 views

For which values of $a$ the matrix is diagonalizable

Given the following matrix: $$B=\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & a^2 \\ 1 & 1 & 0 \end{bmatrix}$$ I tried to find for which values of $a$, the matrix $B$ is diagonalizable. ...
3
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1answer
66 views

How to generate $2\times 2$ matrix with integer entries and both eigenvalues inside unit disk

I am looking for a method to generate $2\times 2$ matrices whose all elements are integers and whose both eigenvalues lie inside the unit disk. I realise that I could do so via similar matrices, i.e. ...
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Does it always hold, that the product of the eigenvalues of a matrix is it's determinant?

I know that this relation holds in several cases but I'm not sure about the full scope of it. So are there any cases when it doesn't hold?
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For which $n, k$ is $S_{n,k}$ a basis? Fun algebra problem

Here it is a nice algebra problem I had some fun with Let $V$ be a vector space over $\mathbb R$ of finite dimension $\dim V = n$. Let $v = \{ v_1, \dots, v_n\}$ be a basis for $V$. Let $$S_{n,k} = ...
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1answer
27 views

Matrix Inversion acceptable Condition Numbers

When considering matrix inversion it is worth while worrying about the condition number of the matrix you wish to invert. Matrices that are poorly conditioned can often create inaccurate results. This ...
3
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1answer
63 views

Eigenvalues of a nxn matrix without calculations [duplicate]

I have a question about the following matrix: $$ \begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \\ \end{bmatrix} $$ Find the ...
0
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1answer
13 views

Generalized eigenvalue problem for symmetric, low rank matrix

I'd like to solve a generalized eigenvalue problem of the form: $$\mathrm{A}x = \lambda \mathrm{B}x$$ $$s.t. x_i^T\mathrm{B}x_i=1.$$ Where $\mathrm{A}$ and $\mathrm{B}$ are symmetric but low-rank ...
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1answer
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System of differential equations, pure imaginary eigenvalues, show that the trajectory is an ellipse.

I am stuck at the last part of a proof. When you have the system of equations: $x'=Ax$ $$ A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ ...
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56 views

If the $det(A)=0$ why does the matrix $A$ have an eigenvector?

If the $det(A)=0$ why does the matrix $A$ have an eigenvector? Explain why there is a basis $B$ in $R^n$ so that the matrix $[A]_B$ has the zero vector as its first column I know that if the ...
3
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2answers
43 views

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear transformation that has eigenvalues $\lambda = -1,0,2$. Find the eigenvalues of $f^2$

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear transformation that has eigenvalues $\lambda = -1,0,2$. Find the eigenvalues of $f^2$. Shouldn't the answer be like this: If $x$ ...
0
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1answer
14 views

Geometric multiplicity of eigenvalues in projection matrix.

Given an $n \times n$ matrix with rank $m$, we can know that the algebraic multiplicity of the eigenvalues of such matrix is: for eigenvalue$=1$ $a.m=m$; for eigenvalue$=0$ $a.m=n-m$. However, is it ...
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2answers
49 views

Polynomials over finite field

I tried to calculate the characteristic polynomial of a 4x4 matrix over the finite field with two elements. I got two results: $x^4+x^3+x+1$ and $(x+1)^3$. First I thought that this must be an error, ...