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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Does negative definiteness imply anything about ALL principal minors?

Unfortunately I haven't received any response for my previous question, so I'm trying to solve it in a different way. I know that iff matrix $H$ is negative definite, its leading principal minors ...
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1answer
19 views

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ with characteristic polynomial $-\lambda(\lambda-3)^2$ and $f(1,0,0)=(1,0,-1)$ and $f(0,1,0)=(2,3,1)$

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ be a diagonalizable endomorphism with characteristic polynomial $-\lambda(\lambda-3)^2$ such that $f(1,0,0)=(1,0,-1)$ and $f(0,1,0)=(2,3,1)$. Given these data, ...
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1answer
7 views

Spectral gap vs. algebraic connectivity

Can someone please clarify how the spectral gap of a graph relates to its algebraic connectivity (aka Fiedler value) and whether these use the adjacency matrix or laplacian matrix?
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1answer
21 views

Finding the real irrational root of a cubic polynomial?

I just wanted to check if anyone can see a simpler way to solve this. Because I am not looking forward to using the cubic formula to solve it! $$ det(\lambda-AI) = \left| \begin{array}{ccc} \lambda + ...
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2answers
44 views

Finding Eigenvalues of given linear operator

Find the eigenvalues and the eigenvectors of the linear operator $T:C^\infty(0, 1)\to C^\infty(0, 1)$ $T(f)(x) = \frac{f'(x)}{x}, x \in (0,1) $ Using the definition : $TF = \lambda F \iff ...
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1answer
33 views

How to bound the biggest eigenvalue of $\sum_{i=1}^{n}x_ix_i^T$?

My question is to bound the biggest eigenvalue of $A=\sum_{i=1}^{n}x_ix_i^T$, where $x_i\in\mathbb{R}^d$ is a column vector. My idea is, to bound the biggest eigenvalue of $A$, i.e. $\|A\|_2$. I can ...
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20 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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2answers
53 views

Can a matrix have eigenvalue with infinite multiplicity?

Suppose we have matrix of the form $$ A= \begin{bmatrix} a & -1 \\ 0 & a \\ \end{bmatrix} $$ and we would like to analyze its diagonalizability. By taking the ...
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5 views

What is a basic description of how diffusion mapping works?

I have been trying to get a basic understanding of diffusion mapping, and I think I understand the concept, but I am having trouble understanding the math behind it (I have knowledge of advanced math ...
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2answers
25 views

Eigenvalues of a transpose multiplication

Say I have a matrix $\mathbf B \in \mathbb R^{m\times n}$. Is it correct to say that the eigenvalues of $\mathbf B^T\cdot\mathbf B$ are always positive?
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29 views

does a closed form solution exist for this equation?

I have a cost function $J$, which depends on a projection matrix $W$, which is unknown. When I get the partial derivative $\frac{\partial J}{\partial W}$ the equation is: $\frac{\partial J}{\partial ...
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1answer
29 views

Eigen value system? solution

I have the following system. $AW = \lambda B W$ Where $A,B,W$ are matrices and $\lambda$ is a scalar. The values of $A,B$ and $\lambda$ are known. $B$ is invertible. This is a solution to an ...
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1answer
34 views

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
45 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 ...
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1answer
53 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 ...
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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
44 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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0answers
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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2answers
31 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
23 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
32 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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10 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
27 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
41 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
24 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 ...
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1answer
25 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
30 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
26 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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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
27 views

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 ...
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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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2answers
26 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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0answers
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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24 views

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
20 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 ...
0
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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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3answers
42 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 = ...
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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
41 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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2answers
31 views

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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0answers
14 views

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 ...
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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
58 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 ...