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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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Why is normalisation of a Laplacian matrix defined as $L - I$?

I have a piece of code that computes the Laplacian matrix using $I - D^{-\frac{1}{2}} A D^{-\frac{1}{2}}$, that is, it computes the symmetric normalised Laplacian. This Laplacian is not defined when $...
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Finding the characteristic polynomial of the $T: \mathcal{M}_{n} (\mathbb{R}) \to \mathcal{M}_{n} (\mathbb{R})$ given by $T(M)=M^{\text{tr}}$

Here's a problem from Larry Smith's Linear Algebra textbook: Let $\mathcal{M}_{n} (\mathbb{R})$ be the set of real matrices of order $n \times n$. Let $T: \mathcal{M}_{n} (\mathbb{R}) \to \...
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Question of the Cholesky decomposition of symmetric positive definite matrix

This is a exercise on my numerical analysis textbook: Suppose $\mathbf A$ is a positive-definite symmetric matrix, and the Cholesky decomposition is of the form $\mathbf {A} =\mathbf {LL}^{*}$, ...
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Lower bound of self-adjoint operator

For a self-adjoint operator $H$, and a nonzero function $u\in H^1(\mathbb R^n)$, if I have $$ \langle H(u), u \rangle_{L^2} < 0 , ~~~~ H|_{u^\bot}\ge 0, $$ then define a new operator $L$, about $...
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Computing Eigenvalues of a large matrix

Let's say a matrix M is composed of: \begin{bmatrix} A & B \\ C & D \end{bmatrix} where $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{m \times n},$ ...
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Rank-1 modification of correlation matrix

I inherited a problem at work and have trouble wrapping my head around it. It doesn't help I'm insufficiently sophisticated with linalg. Help with or pointing to appropriate literature will be much ...
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Why does invertibility of the matrix $A-\lambda I$ imply that $\lambda$ is not an eigenvalue?

Is it because invertibility of the matrix $A-\lambda I$ would imply that $A-\lambda I$ has a trivial null space, and thus there are no solutions to the equation $Ax-\lambda I = 0$ (other than the ...
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The eigenspace of $\lambda$ is simply the null space of the matrix $A-\lambda I$. Is that correct?

Would it be correct to say that the eigenspace corresponding to an eigenvalue $\lambda$ for a matrix $A$ is simply the null space of the matrix $A-\lambda I$? My justification is that the eigenspace ...
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3x3 matrix Eigenvector values does not match

I'm trying to solve a matrix $Q$ (as shown in the SCREENSHOT) to find its eigenvectors. The solutions are provided in the book directly and I was trying to solve it by hand but I cannot match my ...
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Derivative of eigenvalue with respect to a constant

I am having trouble wrapping my mind against a simple problem: Suppose we have the following eigenvector equation for $A\in\mathbb{R}^{n\times n}$ and $\alpha \in \mathbb{R}$. $$ \left(\alpha A\right)...
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Let $T$ be a linear operator on a finite-dimensional vector space, $f$ is any polynomial, find out the relation between eigenvalues of $f(T)$ and $T$.

The question: Let $T$ be a linear operator on a finite-dimensional vector space over an algebraically closed field $F$. Let $f$ be a polynomial over $F$. Prove that $c$ is a characteristic value of $...
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Show that a 2x2 matrix A with complex eigenvalues has a special similarity

My question is: if $A$ is a 2×2 matrix, with complex eigenvalues $(a±bi)$, show that there is a real matrix $P$ for which $P^{−1}AP = \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$...
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1answer
32 views

When do the eigenvectors of a Laplacian matrix form a basis?

Eigenvectors do not always form a basis. When do the eigenvectors of a Laplacian matrix form a basis? When the associated adjacency matrix is symmetric? Why?
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81 views

If $AB=-BA$ then do $A$ and $B$ share a common eigenvector?

I know that for two matrices $A$ and $B$ if $AB = BA$ then they share a common eigenvector. Even in general, for any $k$ if $AB - BA = kB$ then they have a common eigenvector. But what about if $AB=...
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Finding rank-$1$ matrix

Let $$S = \frac{1}{12} \begin{pmatrix} 1 & 10 & 1 \\ 5 & 2 & 5\\ 1 & 2 & 9\end{pmatrix}$$ Find a rank-$1$ matrix $R$ so that $$ M = S + R $$ will have the same eigenvalues as $...
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46 views

Eigenvalues and eigenfunctions of an integral operator

Let $T$ be an integral operator with kernel $K(x,y)=|x-y|$ on $L^2(-1,1)$. How can we find the eigenfunctions and eigenvalues of $T$? Even though I am not sure whether the following arguments are ...
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4answers
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Is there a eigenvalue equal to 0 if determinant is equal to 0?

According to theorem the multiplication of all eigenvalues is equal to the determinant, so if one of them equals 0 the determinant is always 0. But is it true for the opposite statement? If ...
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Finding the eigenvalues of $A=\left(\begin{smallmatrix} a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & a \\ \end{smallmatrix}\right)$

I would like to calculate the eigenvalues of the following matrix $A$, but the factorization of the characteristic polynomial does not seem to be easy to compute. $A=\pmatrix{ a & 1 & 1 \\ 1 &...
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1answer
29 views

Why doesn't inverse iteration always converge towards the eigenvector with the smallest eigenvalue?

Here's my reasoning. Power iteration converges towards the eigenvector with the largest eigenvalue. Inverse iteration is power iteration using the matrix $(A-\mu I)^{-1}$. The eigenvalues of this ...
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How to evaluate the accuray of quadratic eigenvalue problem (QEP)?

When solving the QEP, we transform it into a GEP and then use qz algorithm to handle it. But there are several formulations of GEP, how to evaluate the accuracy and stability of the solution? I ...
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Is my reasoning here correct? [duplicate]

Let's say that we have a matrix $A \in M_n{\mathbb{C}}$ such that $A^2=A-I_n$. Now,I want to see what I can say about its eigenvalues. I think that they are roots of the polynomial equation $x^2-x+1=0$...
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Hyperbolic fixed point of ODE

Suppose we have a linear autonomous two dimensional ODE: \begin{equation} \frac{dx}{dt} = Ax \end{equation} for some matrix $A \in \mathbb{R}^{2 \times 2}$. Now we say a system is hyperbolic if the ...
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Eigenvalues of dense vs. sparse stochastic matrix

For a stochastic matrix $A$, it is known that the maximum eigenvalue is $1$ and that each eigenvalue $\lambda_i$ satisfies the inequality: $|\lambda_i| \leq 1$. Given a dense (and possibly symmetric)...
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Show that $S+T$ has all eigen values non-negative.

Let $V$ be a finite dimensional vector space and $S,T\in \mathcal L(V)$ where $\mathcal L(V)$ denotes the space of linear operators. If $S,T$ are self-adjoint and have all eigen values positive ...
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1answer
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What is wrong with my argument using of Collatz–Wielandt formula?

If $A$ is a positive square matrix, then the Collatz–Wielandt implies that $\min_{𝑖=1,…,𝑛;𝑦𝑖\neq 0}\frac{(Ay)_i}{y_i}≤𝑟≤\max_{𝑖=1,…,𝑛;𝑦𝑖\neq 0}\frac{(Ay)_i}{y_i}$, Where $r$ is the largest ...
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56 views

How is the Laplace Transform a Change of basis?

This question is primarily based on the following answer's way of reasoning, https://math.stackexchange.com/a/2156002/525644 If you want to write a new answer to the question; "How is the Laplace ...
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How to find eigenvalue trajectories under perturbation

Let $A$ be an $n\times n$ diagonalizable matrix, and given a $n \times n$ perturbation matrix $P$. The perturbed matrix $$ B(t) = A + tP, $$ where $t$ evolves in small steps from 0 to 1. I would ...
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3answers
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Matrix such that $A^n=aA$

Let $A\in M_n(\mathbb{C})$ be a matrix such that $A^n=aA$,where $a\in \mathbb{R}-\{0,1\}$. I wanted to find $A$'s eigenvalues and I thought that they are the roots of the polynomial equation $x^n=ax$. ...
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Is the preimage of a simple root connected?

Consider a matrix $A(p) \in \mathbb{R}^{N \times N}$ that depends linearly on a set of real nonnegative parameters $p =[p_1,\dots,p_n], p_j \geq 0$. Let $A$ be nonnegative irreducible for any ...
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Relation between Eigenvalues of a matrix after Row swap

If i have a matrix like \begin{align*} M= \begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 4&-17&8 \end{pmatrix} \end{align*} and merely exchange rows $R_1$ and $R_3$, so i have \...
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24 views

Derivation of eigenvalues of a symmetric matrix

Given a diagonal matrix $D$, we symmetrize matrix $W$ by the following transformation: $S \equiv D^{1/2} W D^{-1/2} $ Now, $S$ is a symmetric matrix which is diagonalizable as $S = X \Lambda X^T$ ...
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35 views

Find the Matrix by given information [closed]

Can we obtain a matrix if we have its eigenvalues, algebraic multiplicity and geometric multiplicity of each eigenvalue ?
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31 views

How to find the rotation vector by deriving the final vector with respect to the displacement?

My understanding of a rotation of a vector can be done by using a 2D rotation matrix as shown below, $R(\theta )=\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{...
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Prove the relationship between the Frobenius-Perron eigenvectors of two positive matrices [closed]

Let $A$ and $B$ be two matrices, where $A \geq B \geq 0$. $\lambda_0(A)$ and $\lambda_0(B)$ represents their Frobenius-Perron eigenvalue. The expression $A \geq B$ means that $\forall\ i,j, \ a_{ij} \...
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Distribution of Singular Values of Subunitary Matrix

Let $U$ be a random $n \times n$ unitary matrix (w.r.t. the Haar measure) and let $M$ be a $k \times l$ submatrix. What is the distribution of the singular values of $M$?
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$\lambda$-eigenspace from matrix with row of zeroes

I am currently revising for my exam on Linear Algebra, and have reached a stump in a question regarding a $\lambda$-eigenspace taken from a $2\times2$ matrix with the bottom row consisting of zeroes. ...
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44 views

If sum of traces of matrices at k-th power is 0, eigenvalues=0?

Given $$A_1^k + A_2^k + \cdots + A_m^k = 0, \qquad \forall k \in \mathbb N^+$$ then $$\mbox{Tr}(A_1^k) + \mbox{Tr}(A_2^k) + \cdots + \mbox{Tr}(A_m^k) = 0$$ where $A_1, A_2, \dots, A_m$ are $n\...
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Must a square root of a diagonal matrix with distinct eigenvalues be diagonal? [duplicate]

Let $D$ be the diagonal matrix and $A^2=D$ So far I've taken 2 approaches, the first being a direct algebraic approach. For a $2 \times 2$ matrix this reduces to $$ \left( \begin{matrix} a_{11} ...
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1answer
26 views

Solving linear vector differential equations with repeated eigenvalue

http://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx This gives an example of how to solve such systems. But I have a problem. what if the eigenspace due to an eigenvalue has dimension ...
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Two proof questions about eigenvectors

a. How do I prove that if $\lambda^2$ is an eigenvalues of $A^2$ then at least $\lambda$ or $-\lambda$ is an eigenvalue of $A$. b. I want to show that an invertible linear transoformation $T$ and its ...
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1answer
66 views

Determine the smallest dimension for eigenspace

Let $A \in \mathbb{R}^{n \times n}$ where $k_i \in \mathbb{R}^n, i = 1,2, \cdots , n$ are column vectors of $A$ and satisfy the following condition $k_i = (i+2)k_{i+2}, i= 1,2, \cdots, n-2$ ...
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1answer
47 views

Prove that a 3x3 matrix always has an eigenvector in $\mathbb R^3$ [duplicate]

I am trying to prove this statement. I am assuming that if a 3x3 matrix always has an eigenvector, then it also always has an eigenvalue. I tried to prove this looking at a general 3x3 case and trying ...
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2answers
64 views

If $T$ is an invertible linear transformation and $\vec{v}$ is an eigenvector of $T$, then $\vec{v}$ is an eigenvector of $T^{-1}$

I saw there is a proof for invertible matrices, but I don't know how to put this mathematically for a transformation. How do I prove an invertible linear transformation has the same eigenvectors as ...
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How do I compute the eigenvectors and general eigenvector of this matrix for $\lambda = 3$?

Let $$A =\begin{pmatrix} 177& 548& 271& -548& -356\\ 19& 63& 14& -79& -23\\ 8& 24& 17& -20& -20\\ 42& 132& 55& -141& -76\\ 56& 176&...
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Trouble understand proof of spectral theorem

I'm reading through this proof of the real spectral theorem. I don't understand the last line of "lucky fact 2" - why must $\overrightarrow{u}$ have been listed in the $v_{i}$?
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Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs

Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
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1answer
73 views

Why is $k$ constant here?

$$k=\frac{f^{T}Be}{f^{T}e}\\ A=-B^{-1}L$$ e is the eigenvector of A and f is the eigenvector of L. L is a Laplacian matrix of an undirected graph (real symmetric, singular & positive ...
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1answer
23 views

Trace and Det of Laplacian on the rectangle

I consider the eigenvalue problem $\Delta \varphi = \lambda \varphi$ with the Dirichlet boundary condition $\varphi|_{ \partial \Omega}=0$ on the rectangle $\Omega= [0,l] \times [0,m]$. By using ...
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1answer
48 views

Name of matrix whose eigenvalues are all conjugate pairs

I've read that "A matrix is positive definite if it’s symmetric and all its eigenvalues are positive" As the title implies...is there a name for a square matrix whose eigenvalues are all conjugate ...
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Eigenvalue problems for elliptic operators on unbounded domains

Assuming $L$ is symmetric, elliptic second order differential operator, I want to to know about solutions to $$ -Lu = \lambda u \quad \text{in } \mathbf{R}^n.$$ Due to the unboundedness of the ...