# 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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### What are the eigenvalues and eigenvectors of $A^2 - 3A + 4I$, given the eigenvalues and eigenvectors of $A$?

Let eigenvalues of $2 \times 2$ matrix $A$ be $1,-2$ and eigenvectors be $x_1$ & $x_2$ respectively. Then eigenvalues and eigenvectors of $A^2-3A+4I$ would be? We know that eigenvalues can be ...
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### Show that $A$ is diagonalizable if and only if $\sum_{\lambda \in \operatorname{spec}_{F}(A)}V_{\lambda} =V$

Let $V_\lambda$ be the eigenspace that corresponds to $\lambda$ and $$\operatorname{spec}_{F}(A) = \{\lambda \in F: V_{\lambda} \neq\{0\}\}$$ I'm asked to show that $A$ is diagonalizable if and only ...
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### Orthogonality of Eigen vectors for the summation of matrices

If a matrix, let's say $A$, can be decomposed into $A_1$ and $A_2$, and the eigenvectors of $A_1$ and $A_2$ are orthogonal. Does it guarantees that the Eigenvectors of $A$ would also be orthogonal?
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### Eigenvalues of $BB^T$ where $B$ is the Incidence Matrix

Let $I_n, J_n \in M_n(\mathbb{R})$ where $I_n$ is the identity and $(J_n)_{ij} = 1$ for all $i,j$. (1) Calculate the determinant, eigenvalues and eigenvectors of $I_n$ and $J_n$. (2) ...
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### Intuition behind Eigenvalue solution matrix

I've been watching the excellent course by 3Blue1Brown on Linear Algebra which is oriented towards giving students intuition into Linear Algebra concepts. I am trying to find an intuitive way to ...
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### $T: \mathbb{R}^n \to \mathbb{R}^n$ linear with $n > 1$. Then there is $M$ with $\dim M = 2$ and $T(M) \subset M$.

Let $T: \mathbb{R}^n \to \mathbb{R}^n$ be a linear map, with $n > 1$. Prove that there is a subspace $M \subset \mathbb{R}^n$, with $\dim M = 2$ such that $T(M) \subset M$. This question is from ...
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### Similarity Invariance of Trace for Matrix Multiplication

From wiki, I know that for a square matrix A and any invertible matrix P of the same dimensions, the following would hold: $$tr(P^{-1} A P) = tr(A)$$ In my case, A is a density matrix. The question ...
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### Finding the Eigenvectors

Consider the matrix given below: $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}.$ The eigenvalues for this matrix are $\dfrac{1+\sqrt 5}{2},\dfrac{1-\sqrt 5}{2}.$ I am facing trouble ...
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### Question regarding the multiplicity of eigenvalues of a matrix polynomial .

$\mathbf {The \ Problem \ is}:$ If $\operatorname p(x)$ is a polynomial over an algebraically closed field $\mathbb F[x]$ and let $A$ be an $n×n$ square matrix, then it is known that every eigenvalue ...
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### In a primitive, stochastic matrix, the 2nd largest eigenvalue depends on which property of the elements of matrix?

We know the largest eigenvalue of a primitive, stochastic matrix is 1. The 2nd largest eigenvalue is strictly smaller than 1. But can we get more information of 2nd largest eigenvalue from the ...
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### Conditions for two eigenvectors of $AA$ to be eigenvectors of $A$

Let $V$ be a $d$-dimensional real vector space with inner product $\langle\cdot{,}\cdot\rangle$, and suppose $A$ is a symmetric linear map $V\to V$. Then, by the spectral theorem, $A$ has an ...
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### Is every eigenvector of AA an eigenvector of A?

Let $V$ be a (finite-dimensional) vector space and $A \colon V \to V$ a linear map. Is it true that, if $v$ is an eigenvector of $A\circ A$, then $v$ is an eigenvector of $A$? I know the converse ...
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### Find determinant of $B$ when $B=2A+A^{-1}-I$

Let $A$ be a $4\times 4$ real matrix with eigenvalues $1,\ -1,\ 2,\ -2$. If $$B=2A+A^{-1}-I$$ then determinant of $B$ is? We are provided with eigenvalues then from Cayley Hamilton theorem, $A$ ...
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### Solution of $AX+XA=B$ through eigenvectors of $A$

I see that Matlab uses the spectral decomposition to solve the continuous Lyapunov equation $$AX+XA=B$$ The formula they use for positive definite $A$ with matrix of eigenvectors $U$ and column ...
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### Prove that vector is eigenvector.

Let $A$ be a matrix of size $5$ (real values) and real $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, such that: vector $(1,1,1,1,1)$ is eigenvector of $A$ corresponding to eigenvalue $\lambda_{1}$ ...
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### Are the minimum of two symmetric matrices closer to each other than the two matrices themselves?

Consider any two real symmetric matrices $A,B\in\mathbb{S}^n$ with spectral decompositions: $$A=U_A D_A U_A',\quad B=U_BD_BU_B'$$ Now, let $$A^-\equiv U_A\min(D_A,0)U_A',\quad B^-=U_B\min(D_B,0)U_B'$$ ...
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### Sample from distribution given by sum of eigenvalues of symmetric matrix

I'm interested in (efficiently) sampling from the following density: $$p(X) \propto \exp\Big(\sum_{i=1}^n \lambda_i(X)\Big),\quad\textrm{on}\ \{X \in \textrm{Sym}(n) : \lVert X \rVert_F \le 1\},$$ ...
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### spectral radius of products of symmetric positive definite matrices

I have matrices $A, B, C, D \in \mathbb R^{n\times n}$ which are all symmetric positive (semi-)definite. If I have that \begin{align} \rho(AB) &< \gamma^2, \\ \rho(CD) &< \gamma^2, \...
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### Algebraic formula of the pseudoinverse (Moore-Penrose) of symmetric positive semidefinite matrixes

I was reading the Wikipedia page about the Pseudoinverse or Moore-Penrose inverse, where they say that, given a generic matrix $A \in R^{nxm}$, if the matrix is full rank (i.e. rank=$min\{n,m\}$), ...
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### Eigen Decomposition Check

I am following the wiki entry on eigen dicomposition with the following matrix: $$A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$$ I wish to find a diagonalizing matrix T.S. T^...
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### Methods for Definiteness of a matrix are giving different results

I've found several different sets of rules to check for definitness. They are: Determinant of all upper-left submatrices, shown here Check the sign of the eigenvalues of A, which is the definition ...
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### Is it possible to go from Eigenvectors to the original nxn matrix from which it was decomposed?

I understand how eigenvectors can be decomposed from a matrix. Now let's say, I have a set of eigenvectors in hand, is it possible to compute the original matrix from it? Also, does it make sense to ...
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### How to find eigenvalues of the matrix

This is a question from our end-semester exam: How to find the eigenvalues of the given matrix: M=\begin{bmatrix} 5,1,1,1,1,1\\ 1,5,1,1,1,1\\ 1,1,5,1,1,1\\ 1,1,1,5,1,1\\ 1,1,1,1,4,0\\ 1,1,1,1,0,4\\...
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### Are there any nice properties we can talk about after using PCA to transform data to a new basis?

I'm not talking about "compressing" data here....assume we kept all the eigenvector basis after performing PCA, and then used them to transform our data into a new basis of the same dimension as the ...
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### Eigenvalues of an outer product

Let $w, v \in \mathbb{R}^d$. What is known about the eigenvalues of the outer product matrix $vw^{\top} + wv^{\top}$?
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### Minimizing product of a vector with a symmetric matrix

Is the minimum of ${\left\lVert{x^TA}\right\lVert}$, where $x \in \mathbb{R}^n$, and $A^{n \times n}$ is a symmetric real matrix, related to the smallest eigenvalue of $A$? I read about Rayleigh ...
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### Matrix diagonalization. Is $A = PDP^{-1} = P^{-1}DP$?

Diagonalization of a square matrix $A$ consists in finding matrices $P$ and $D$ such that $A=PD P^{-1}$ where $P$ is a matrix composed of the eigenvectors of $A$, $D$ is the diagonal matrix ...
### Eigenvalue of $A-aI_3$
Question: Let $A=\begin{pmatrix} a+1 & 1 & 1 \\ 1 & a+1 & 1 \\ 1 & 1 & a+1\end{pmatrix}$. Show that $A-aI_3$ has eigenvalue of 3. Also find eigenvector. My thinking: I ...