# 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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### Mathematical problem related to spectral method for time evolution

I have a mathematical problem related to spectral method for time evolution in QM. Considering the standard evolution for a generic quantum state $\psi(t) \in \mathbb{C}^N$ which is not express in a ...
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### BA diagonalisable show that AB diagonalisable ( A and B are not square) [duplicate]

let A $\in \mathcal{M}_{4,3}(\mathbb{R})$ and B $\in \mathcal{M}_{3,4}(\mathbb{R})$ such as BA$= \begin{pmatrix} 0&1&1 \\ 1&0&1 \\ 1&1&0 \end{pmatrix}$ show that AB is ...
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### Eigenvector of the product of a rank-one matrix and an arbitrary matrix

Suppose $A$ is a rank-one (real) matrix ($n \times n$) of the form $u v^t$ and $B$ is an any real square matrix ($n \times n$) with a dominant eigenvalue. Can it be shown that $A$ and $AB$ have the ...
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### Is it true that the dimension of an eigenspace of a square matrix is at most the multiplicity of the corresponding eigenvalue?

By "the multiplicity of the corresponding eigenvalue", I mean the multiplicity of the eigenvalue as a root of the characteristic polynomial $p(x)=\det(xI-A)$, i. e. the maximum integer $k$ ...
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### Help approximating the leading eigenvalue of this Jacobian

I have a system of 4 ordinary differential equations with positive parameters, which has two stable fixed points. The Jacobian matrix evaluated at the first equilibrium leads easily to finding the ...
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### Find the axis of reflection which is described a matrix

I need help with the following example. Find the axis of reflection which is described by a matrix $\frac{1}{10}\begin{pmatrix} 3 & 1 \\ -1 & 3 \end{pmatrix}$ Should I use eigenvectors ...
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### Eigenvalues equation and differential equations [closed]

Can we express the Hermite's differential equation $$\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+2ny=0$$ as an eigenvalue equation? If possible, then how? Can someone elucidate a little bit?
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### Proving that $a_kA^k+a_{k-1}A^{k-1}+\cdots +a_1A+a_0I = 0_n$

I've been studying linear algebra and came across this question that I have no general idea how to solve. The question is as follows. Let $A$ be an $n×n$ diagonalizable matrix with distinct ...
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### Find eigenvalues of $I - uv^T$

I want to show that the eigenvalues of $I - uv^T$, where $u,v \in \mathbb{R}^n$ are given by 1 with multiplicity $n-1$ and $1-v^Tu$ with multiplicity 1. I have tried setting up the eigenvalue equation ...
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### Eigenvalues when perturbed along anti-diagonal.

Given a $n \times n$ matrix $A$ with eigenvalues $\lambda_k$ for $k = 1, 2, \dots, n$ we know the relationship between $\{\lambda_k\}$ and the eigenvalues of $A + t I$ where $I$ is the identity matrix ...
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### Change of eigenvalues and eigenvectors caused by matrix expansion

I have a set of $k$ data pieces, each data piece is an $n$-d vector. Each data piece here could be an image block or a piece of music. The dataset could be expressed as an $n\times k$ matrix $A$, that ...
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### Comparing two Diagonalisable matrices with identical specific features

Hello, I was wondering if anyone could help me with this Linear Algebra question that I'm stuck on. So far this is what I've worked out: I know that if a Matrix is Diagonalisable then it's diagonal ...
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### Linear map questions in association with eigenvalues, eigenspace and diagonalisablility.

Hi I was wondering if anyone could help me on this question I have become stuck on. This is what I know so far: For c) I know in general the algebraic multiplicity and geometric multiplicity of an ...
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### Let $T:V \rightarrow V$ such that $T^2 = \frac{1}{2}T$. Find its characteristic polynomial.

I am used to dealing with transforms such as $T:P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R})$ where $T(ax^2 + bx + c) = (a+b)x^2+cx$. In this case you would just use a basis of $P_2(\mathbb{R})$ and ...
### Geometric multiplicity for non zero eigen values of matrices $AB$ and $BA$.
As lot of information is given in this site about eigen values of $AB$ and $BA$ for square matrices $A$ and $B$. As characteristics polynomial of $AB$ and $BA$ are same so both have same set of eigen ...
Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. Written as: \frac{\partial{u}...