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Questions tagged [eigenvalues-eigenvectors]

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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$V=\mathbb{M}_{n}(\mathbb{F})$, $f\in V^{*}$ with $f(AB)=f(BA),$ exists $\lambda\in\mathbb{F}$ with $f(A)=\lambda\textrm{tr}(A)\,\forall \,A\in V$

Let $V=\mathbb{M}_{n}(\mathbb{F})$ the space of $n\times n$ matrices. Let $f\in V^{*}$ satisfying $f(AB)=f(BA).,$ So, exists unique $\lambda\in\mathbb{F}$ such that $f(A)=\lambda\textrm{tr}(A)\,\...
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Laub Matrix Analysis Theorem 9.15

From Alan J Laub, Matrix Analysis For Scientists and Engineers, 2004, p 79 Theorem 9.15. Let $A \in \mathbb{C}^{n\times n} $ have distinct eigenvalues $\lambda_1, \ldots, \lambda_n$ and let the ...
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What is the explicit formula for the general term of the sequence?

A sequence $\{a_n\}_{n \ge 1}$ is defined recursively by $$a_0 = 1, a_1 = 1$$ $$a_n = 5a_{n-1}-6a_{n-2}, \text{ for } n \ge 2 $$ Find an explicit formula for the general term $A_n$. So, I want to ...
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Linear stability analysis on a simple pendulum

So I have a simple pendulum (rod has no weight, point mass, no frictional forces) and I’m measuring the angle theta from the downward vertical, hence I have the governing equation $$\ddot{\theta}+sin(\...
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Finding demonstration of I Debreu's Theorem

Can someone tell me where i can find on web the demonstrations of first Debreu's theorem? Just to be clear: Given $Q:\mathbb{R}^{n}\rightarrow \mathbb{R}$, with $Q(\bar{x})=\bar{x}^{T}A\bar{x}$ a ...
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Matrix functions reducing eigenspectrum degeneracy?

Standard matrix functions fulfill $spec(f(A))=f(spec(A))$ relation for eigenspectrum, maintaining degeneracy if $f$ is injection. For simplicity we can focus on real symmetric matrices. However, ...
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Does it make sense to write $(T - \lambda I)v$ if $T$ is a linear transformation?

I am reading this page to try to understand Jordan form. $V$ is a finite-dimensional complex vector space, and until now $T$ has always represented an "operator", by which I guess they mean a linear ...
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Don't understand this Sturm - Liouville problem

All the SL problems I have seen before had $ \lambda$ in them, so finding the eigenvalues meant finding the values of $ \lambda$. However I don't know what I am supposed to do in the following: Find ...
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Eigenvalues of A are also eigenvalues of T

Let $V$ be the set of all $n\times n$ matrices over a field $F$. Let $A$ be a fixed element of $V$. Define a linear operator $T$ on $V$ by $T(B)=AB$. I am trying to show that if $\lambda$ is an ...
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Calculate variations of eigenvectors w.r.t input matrix

From the eigen decomposition : $$A = PDP^T$$ I would like to calculate $dP$ w.r.t $dA$. I start like this : $$dA = dPDP^T + PdDP^T + PDdP^T$$ Since we only consider variations in eigenvectors and ...
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Verify that an eigenvector is orthogonal to 2 other eigenvectors

Verify that the eigenvector v3=(4,-2,1) corresponding to the eigenvalue e2=16 is orthogonal to the eigenvectors v1=(1/2,1,0) and v2=(-1/4,0,1) (both) corresponding to eigenvalue e1=-5 All I can ...
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Orthonormal Eigenbasis of the reflection matrix

So this question is in a way more about computation than theory, because I feel pretty confident in the latter but yet can't get the former to work. What I seek is given $\begin{pmatrix} \cos(\...
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Minimization Problem involving PCA

Starting from this equation to estimate: $\DeclareMathOperator{\tr}{Trace}$ $$ X = F^k \Lambda' ^ {k} + e, $$ with $ X$ an $T\times N $ matrix, $ F$ is $T\times k $ and $ \Lambda$ is $k\times k $, ...
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Prove the moment generating function of the random variable is the function of its eigenvalues.

I want to prove that $$ \mathbb{E} \Big\{ \exp\left(-x\lVert{\mathbf{H}}\rVert_F^2\right)\Big\} = \frac{1}{\det(\mathbf{I}_{m,n}+x\mathbf{R})}=\prod_{i=1}^{m\times n}\frac{1}{1+x\lambda_i(\mathbf{R})},...
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A question in SVD

if $A\in \mathbb{R}^{m \times n}$, then SVD of A is $U\Sigma V^T$. I've seen different version of SVD. In the first one, $U\in \mathbb{R}^{m \times m}$, $\Sigma \in \mathbb{R}^{m\times n}$, and $V \in ...
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How many basis vectors are there in an eigenspace of dimension k?

If $T:V\to V$is a linear map and we know that $\lambda$ is an eigenvalue of $T$ and the eigenspace of T wrt $\lambda$ has dimension $k$ then does that mean there are $k$ linearly independent ...
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$AB$ and $BA$ have the same eigenvalue? [duplicate]

If $A,B :V\to V$ linear transformations then $AB$, $BA$ have the same eigenvalues? I can not find counterexample, so i think that they have the same eigenvalues , but I do not why, they are similar ...
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Proving a vector is an eigenvalue given an equation involving the matrix

i am told that v is an eigenvector of A with eigenvalue p. I am to show that p is an eigenvalue of A^3 -4A^2 + I, and find it's eigenvalue. I have first shown that p^3 is an eigenvalue of A^3 by ...
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Finding eigenspace

I have been doing the following exercise problem: Let $T: R^3-> R^3$ be a linear transformation so that: Determine eigenvalues and eigenspace of T. So, I determined that $0$ and $1/2$ are ...
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Finding eigenvalues of a matrix given its characteristic polynomial and the trace and determinant

I am told a matrix A has characteristic polynomial: $(\lambda−1)^3(a\lambda+\lambda^2+b),$ and that $\text {tr}(A)=12,$ and $\det(A) =14.$ I am asked to find the eigenvalues. Is the only to do this ...
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Why is my first eigenvector nearly constant and associated to a very high eigenvalue?

The first eigenvector I obtain after decomposing an N x N symmetric correlation matrix (computed using a custom correlation measure) has very little variability in it. It is also associated to a very ...
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Eigenvalues, and eigenvectors proof for multiplicity [on hold]

We were given this to prove: Let $A$ be a matrix with an eigenvalue $λ$ that has an algebraic multiplicity of $k$, but a geometric multiplicity of $p$ less than $k$, i.e., there are $p$ linearly ...
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Phase Portraits in 3D differential equations

We know that when we have two equations and two variables, there are certain rules that make the phase portrait a saddle, node, etc... based on whether eigenvalues are positive or negative. For ...
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Find a eigenvalues of matrix $A$ without using characteristic polynomial

Let $A$ be the following matrix: $A=\begin{bmatrix} 4&1&-1\\ 2&5& -2\\ 1&1&2\\ \end{bmatrix}$ Find the eigenvalues of $A$ if you know that algebraic multiplicity of ...
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Minimal polynomial of operator with all eigenvalues are 1

I'm trying to understand proof of this lemma. I have next questions: 1)If $\tau$ acts as the identity on the subspace $\mathbb{R}\alpha$ and on the $E/\mathbb{R}\alpha$ doesn't it mean that $\tau$ ...
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How to find eigenvectors and choosing free variable

I have this matrix: $$ \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ and I have to find the eigenvectors of the upper matrix. As ...
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Bound on eigenvalues of matrices of the form $XDX^T$

I encountered this form of a Matrix while analyzing Logistic Regression (It's the Hessian). Let $H = XDX^T$, where $D$ is a positive definite diagonal matrix with maximum diagonal entry as some $c$ ...
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$3$-by-$3$ Real Matrices with Repeated Eigenvalues

The questions I have are as follows. Prove that for $3 \times 3$ matrices with repeated eigenvalues, all eigenvalues are real. Prove that if two eigenvalues of $3 \times 3$ are complex conjugate, ...
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Eigenvalues of a special symmetric matrix

Can somebody help me in finding eigenvalues of the symmetric matrix $ \pmatrix{A & B\\ B & C}$? Here $A$ and $C$ are symmetric matrices of order $n$ and $B$ is a diagonal matrix of order $n$....
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Characteristic polynomial is divisible by $(x-\lambda)^r$

Let $\lambda$ be an eigenvalue of a linear map $T: V \rightarrow V$, where $V$ is a vector space over $\Bbb{C}$. Let $W$ be the associated eigenspace which has dimension $r$. Then we need to prove ...
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Differentiability of largest eigenvalue for a $C^1$ function

I encountered following interesting statement: If $f: [a, b] \to M_n(\mathbb C)$ is a $C^1$ ( $C^1$ in the interior and left/right differentiable over the end points) function over an interval $[a, b]...
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Kernel of a linear transformation $D : \mathbb{C}^{\infty}(\mathbb{R}) \to \mathbb{C}^{\infty}(\mathbb{R})$

Let $D : \mathbb{C}^{\infty}(\mathbb{R}) \to \mathbb{C}^{\infty}(\mathbb{R})$ be the function given by differentiation: $D(f) = f'$. I've shown that $D$ is a linear transformation by using rules for ...
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Eigenvalues of the Product of a Diagonal and a Symmetric Matrix

Let $A \in \mathbb{R}^{n\times n}$ be a symmetric matrix and $D \in \mathbb{R}^{n\times n}$ be a diagonal matrix with positive entries. Prove that the matrix $P:=DA$ has real eigenvalues.
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Order of equation that is used to get the matrix for calculating eigenvalues?

My professor in class has taught this equation as being $det(\lambda I - A)$ mostly with regards to finding the eigenvalues and eigenvectors of liner transformations but I have seen it online as $det(...
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Why does $LA = \Lambda L$ represent $n$ systems of equations of the form ${{\bf{l}}_i}^TA = {\lambda _i}{{\bf{l}}_i}^T$?

Looking at the eigenproblem $LA = \Lambda L$, where $A,\,\;L\;$ and $\Lambda $ are real n-by-n matrices with $L = \left( \begin{array}{l} {{\bf{l}}_1}^T\\ \;\; \vdots \\ {{\bf{l}}_n}^T \end{array} \...
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Condition for product of tow rectangular matirx is diagonalizable?

Let $A$ and $b$ be $m \times n$ matrices, it seems the product $A'B$ is diagonalizable only if $A$ and $B$ share the same left and right singularvectors. Is it true ?. How can I prove it, in case ...
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Eigenvectors and Eigenvalues of a Sum of Specific Lower Rank Matrices

Suppose I have the follow matrix: $X \in \mathbb{R}^{N\times N}$. Now suppose I try to rewrite $X$ as the sum of some lower rank matrices of a specific form. In particular, by: $X = 0.5X_{\text{...
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Is that true that if A - B is positive definite, then A's eigenvalues are larger than B's [duplicate]

Suppose that $A$ and $B$ are two Hermite matrices and that $A - B$ is positive definite. Denote the eigenvalues of these two matrices by $\lambda_1(A) \ge \lambda_2(A) \ge \dots \ge \lambda_n(A)$ and $...
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3x3 Determinant, Solving for Eigenvalues

For my class in dynamical models in biology, we analyze the local stability of steady states of systems of differential equations by taking a linear approximation and finding the Jacobian matrix. ...
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Size of the smallest matrix satisfying $vA^k=v$ and $vA^i \neq v$ for all $i=1\dots k-1$

Suppose that the following holds for a rational matrix $A$ and a vector $v$. \begin{align*} vA &\neq v \\ vA^2 &\neq v \\ &\vdots \\ vA^{k-1} &\neq v \\ vA^k &= v \end{align*} ...
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Show that if the eigenvalues of a real matrix are not real, then the matrix cannot be symmetric

Question for the problem I have not been able to make any progress in this problem and would appreciate any help if possible. I have been able to prove that eigenvalues of a symmetric matrix are ...
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Finding the trace & determinant of a linear operator which takes any square matrix as an input.

We define $T(X) = AX$ with $A$ & $X$ square matrices of size $n$ with complex entries. First, write down all the eigenvalues (with their respective algebraic multiplicities) of $T$. Use this to ...
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Eigenvectors and Eigenvalues - cubic equation?

So, I have the fact that : Av = λv Where λ is the eigenvalue. I tried substituting in v=A^-1*v* λ to the equation but this didn't get me anywhere. Any help???
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The eigenvalues of $\begin{pmatrix}0&A\\A^*&0\\ \end{pmatrix}$ are the singular values of $A$ along with the negative signs.

The eigenvalues of $\begin{pmatrix}0&A\\A^*&0\\ \end{pmatrix}$ are the singular values of $A$ along with the negative signs. Here $A$ is an $n \times n$ matrix, has $n$ singular values. Here ...
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Finding the eigenvalues of a linear transformation which takes inputs from the set of all $n\times n$ matrices.

We define $T(X) = AX - XB$ for fixed $A,B$. We allow $X$ to be any matrix in $M_n(F)$. Write down all the eigenvalues of $T$ in terms of the eigenvalues of $A$ and $B$. I think I saw another ...
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Interpretation of signs and magnitudes of eigenvalues of Hessian

Suppose I have a 50-dimensional field. I compute the Hessian matrix at a stationary point and find 40 negative eigenvalues + 10 positive ones. Can I conclude that the point is "mostly" a maximum with ...
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If $2\times 2$ matrix $A$ is symmetric, why can we find a pair of orthonormal eigenvectors $q_1=(x_1,-y_1)^T$, and $q_2=(y_1,x_1)^T$?

I am okay with the fact that we can find a pair of orthonormal eigenvectors, but how do we know that we can find some of the form $q_1=(x_1,-y_1)^T$, and $q_2=(y_1,x_1)^T$? NOTE: I'm not sure if we ...
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Is $\|\lambda\|_3 << \|\lambda\|_2$ for all sub-matrices of a random matrix?

Let $A\in\{0,1\}^{n\times n}$ be a random, symmetric matrix, in which each upper triangular entry is sampled iid. from Bernoulli($p$). Let $\lambda=(\lambda_1, \dots, \lambda_n)$ be the vector of ...
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Eigenvalues relation between matrices that have “similar” rows

Let $A$ be an $n\times n$ matrix whose eigenvalues have all negative real part. Then we construct the matrix $B$ by multiplying each row of $A$ with a positive number, i.e. $(B)_{ij} = a_i (A)_{ij}$ ...
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Geometric intuition of Jacobi rotation

I am studying jacobi rotation Could you explain the geometric intuition for what the jacobi rotation does in calculating the Eigen values of a matrix.