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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Do we need unit length eigenvectors for the process of matrix diagonalization?

I diagonalized a 2 by 2 matrix recently, and I did the usual process: Find eigenvectors of the matrix $A$, put those column vectors together side by side to form a new matrix $S$, then compute $S^{-1}...
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First eigenvalue of laplacian on a cylinder

I am looking for a reference for the value of the first eigenvalue of the laplacian on the right cylinder $$C(r,h) = \{ (r \cos \theta, r \sin \theta, z) : \theta \in [0, 2\pi], z \in [0,h] \}$$ ...
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30 views

Finding basis of eigenspace with eigenvalues

Disclaimer: My English vocabulary for the few nouns used here are google results, so no idea if they are the correctly translated equivalent. Currently sitting on following problem: I have ...
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33 views

Proof that $A$ is similar to $B$

The matrices are defined with $A = \left(\begin{array}{rrrr} -1&2&-3&-1\\% 8&-7&12&4\\% 6&-6&10&3\\% 2&-2&3&2\\% \end{array}\right)$ and $B= \left(\...
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Derivative of eigenvalue of matrix with respect to its elements

Assuming that matrix $A$ is positive semidefinite and that $\lambda$ denotes the eigenvalue, I would like to compute the following gradient $$\nabla_A \lambda(A)$$ I wanted to set this problem up ...
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Continuous dependence of eigenvalues on tangent-space endomorphisms

Consider the following setting: Let $M$ be a smooth manifold and suppose that $T \colon M \to T^{(1,1)}(M)$ is a smooth section, that is, $T_x \in \mathrm{End}(T_xM)$ for every $x \in M$. Moreover, ...
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How to prove two matrices are not similar when the geometric multiplicity of these matrixes are not equal

I have been given two matrices that are defined as follows: $$A = \left(\begin{array}{rrrr} -1&2&-3&-1\\% 8&-7&12&4\\% 6&-6&10&3\\% 2&-2&3&2\\% \end{...
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Numerical stability: cannot unitarily diagonalize normal matrices

I am trying to setup small numerical experiments to see if a unitary matrix can be unitarily diagonalized thanks to the spectral theorem: https://en.wikipedia.org/wiki/Spectral_theorem (see normal ...
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maximum eigenvalue across subsamples

I have an $N$-dimensional vector of data, say $X_{t}$, with $1 \leq t \leq T$. Of this vector $X_{t}$, I want to consider sub-vectors, say $X_{t}^{b}$, which are $m$-dimensional combinations of ...
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Given a matricial expression, the eigenvalues are restricted to the polynomial solutions?

Given a matricial expression, like $A^2-4I=0$, I want to know if it is true that the eigenvalues of $A$ are restricted to the solutions of the polynomial $x^2-4=0$, so $x \in \{-2,2\}$. With Cayley-...
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calculate eigenvalues and determinant of symmetric matrix [on hold]

I have encountered a problem in my exam which I want to find the exact solution. The question is as following: The symmetric matrix $$A_{8\times8}$$ on condition $$A^2=8I$$ and and the sum of the ...
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Calculating Eigenvectors of a Diagonal Matrix

I'm feeling dumb even asking this. But there might be a definition for this somewhat like why $1$ is not a prime number. Therefor this might be the right place to ask this question anyway. Given the ...
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Linear transformations, eigenvalues and basis [on hold]

1) Let $B=《a;c》$ be a basis and let $T$ be a linear transformation that verifies $T(a)=2a+6c; T(c)=-7a+5c$. Find the matrix $[T]BB$ 2) Let $B=《a;c》$ be a basis and let $T$ be a linear ...
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Proving that a symmetric and idempotent matrix has all eigenvalues equal to 1

Let $A^{n \times n}$ be a symmetric and idempotent matrix. $A^t=A$ and $A^2=A$. Prove that $A$ has $n$ eigenvalues that are all equal to 1. I'm having some difficulty proving that. Here's what I've ...
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Number of triangles in $(n,n/2,2\sqrt{n})$-expander graph.

Let $G$ be an $(n,n/2,2\sqrt{n})$ expander graph. That is, the second largest eigenvalue of the adjacency matrix is $2\sqrt{n}$. Let $M$ denote the number of triangles in $G$, I want to show that as $...
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Find $T(1,2,i)$ where $T$ is normal and the eigenvalues are given

I have the following question: Let $V=\mathbb{C}³$ with canonical inner product and let $T\in \mathcal{L}(V)$ satisfing the following properties: i) The only eigenvalues of $T$ are $i$ and $-...
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37 views

Finding a basis of a generalized eigenspace

I have to find a basis for the generalized eigenspace $\ker(A-\operatorname{Id})^3$, where $A\in M_n(C)$ is given by : $$ A=\begin{pmatrix} 1 &1+i& 2&3-i \\ 0 & 1+i & 1 & 2-i\\...
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1answer
35 views

How to find the diagonal matrix when transformation and conversion matrices are known?

I am stuck with finding the diagonal matrix for eigenvalues. Given the matrix T = \begin{bmatrix}6&-1\\2&3\end{bmatrix} and change of basis matrix C= (whose columns are eigenvectors of T) \...
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How to get positional coordinates from distance matrix? (follow-up on previous question)

I have a distance matrix D_ij. I would like to compute the positional coordinates that yield this given distance matrix. I browsed the stack exchange forums and ...
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1answer
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On the bound of eigenvalues for any matrix

I was reading this preprint: https://arxiv.org/abs/1605.00531. The authors mention the following: "It is known that the eigenvalues $z_k$ of any matrix $Z$ are in the rectangle $\text{Re}( z_k) \in \...
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how to uncouple differential equations

Consider the following equations: $$ \dot{u}_i(t)=-\mu_i u_i(t) + \sum_{j\neq i} J_{ij}u_j(t) \quad \text{with }1\leq i\leq N $$ although this is a linear system this is a real pain to solve when $N$ ...
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Given a diagonalizable linear map, find a basis in which the matrix is strictly upper triangular

I have been given the following definition: Let $K$ be a field, $n \in \mathbb{N}$, and $V$ an $n$-dimensional $K$-vector space. Furthermore $\varphi$ is an Endomorphism of $V$, such that there is a ...
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Trace($A$) = Trace($A^2$) = Trace($A^3$)

Suppose $A$ be an $n\times n$ matrix with real eigenvalues such that $\text{trace}(A)=\text{trace}(A^2)=\text{trace}(A^3)$. What can we conclude about the eigenvalues of A? My attempts: Considered $...
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eigenvalue of a giant matrix.can't see the pattern to solve [on hold]

\begin{bmatrix} -1-s&a&b&c&0&d \\a&-s&b&c&d&0\\b&c&-s&d&0&a\\c&d&0&-s&a&b\\d&0&a&b&-s&c\\0&a&...
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3answers
37 views

Finding eigenvalues without doing calculations

Find, without doing any calculations, the eigenvalues of the following linear transformations from $\mathbb{R}^2 \to \mathbb{R}^2$: $A)\quad$ Projection on a straight line that contains $(0,0)$ ...
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1answer
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All eigenvectors of a real symmetric matrix are orthogonal to one another.

I did part of the proof where the eigenvectors corresponds to distinct eignevalues, where at the conclusion we get (λi - λj)xi'xj = 0 which follows that xi'xj = 0, where λi ≠ λj. I want to know how ...
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3answers
70 views

Eigenvalues of symmetric matrix 4x4

I have to calculate the eigenvalue of this symmetric matrix: $\left[ \begin{array}{rrrr} u & u & u & v \\ u & u & v & u \\ u & v & u & u \\ v & u & u & ...
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Relation between two matrix sequences that share the same eigenvalue distribution

Given a sequence $\lbrace A_n \rbrace_n$ of matrices of increasing dimension ($A_n \in M_{d_n}(\mathbb{C})$ with $d_{n+1} > d_n$), we say that the sequence is distributed with respect to the ...
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analytic formula for eigenvalues of symmetric matrices

For a project I am working on, I would like to find symmetric matrices depending on a parameter for which the eigenvalues can be written analytically and the corresponding eigenvectors vary as the ...
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Prove or disprove, all eigenvalues of a real symmetric matrix are non-negative. [closed]

I tried to find an answer for this question, but what I found was a classification of general matrices (i.e. definite, semi-definite and indefinite). I want to know more specifically about symmetric ...
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Faster way to compute eigenvalues of a matrix doing rows operations

I had to find the eigenvalues of this matrix: $\begin{bmatrix} -20 & -25 & -22 \\ 6 & 11 & 12 \\ 5 & 5 & 1 \end{bmatrix}$ I found the characteristic polynomial: $P(\lambda) =...
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Why do we use the zero vector for computing eigenvectors?

While computing the eigenvectors, we use the zero determinant. I know what it states for. However, I can't understand why we do need to have a zero vector, which is stated in the definition of ...
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Prove that there exists an $m \times m$ matrix $X$ such that $AX-XA=C.$

Let $A$ be a real symmetric $m \times m$ matrix with $m$ distinct eigenvalues and $v_1,v_2, \cdots , v_m$ be the corresponding eigenvectors. Let $C$ be an $m \times m$ matrix such that $\left \langle ...
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3answers
366 views

Eigenvalue and similar matrices

if $A$ and $B$ are two $n\times n$ matrices with same eigenvalues such that each eigenvalue has same algebraic and geometric multiplicity. Does $A$ and $B$ are similar? If $A$ is diagnalizable then ...
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Simple eigenvalue

A linear operator $T$ on a separable Hilbert space $H$ is said to be a weighted shift operator if there is some orthogonal basis $\{e_n\}_n$ and weight sequence $\{w_n\}_n$ such that $$Te_n=w_n e_{n+1}...
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Prove that the smallest eigenvalue of a symmetric matrix $A$ is equal to the minimum value of $u^TAu$ where $u$ is a unit vector in $\Bbb R^n$.

Let $D = \{u \in \mathbb{R}^n | \|u\|=1\}$ denote the unit sphere. Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix, and denote its smallest eigenvalue by $\lambda_\min$. Why do we have $$\...
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Existence of $n-1$ dimensional invariant subspace of $V$ over $\mathbb{R}$ given characteristic polynomial has a real root.

$V$ is a finite dimensional vector space over $\mathbb{R}$ with $\dim V \ge 1$ and $\phi \in L(V, V)$ is an endomorphism. Its characteristic polynomial $w_{\phi}(\lambda)$ has a real root. Prove the ...
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Let $A$ be a square $n \times n$ matrix such that $A^2 = 2A$. Show that $A$ has only two distinct eigenvalues.

I have tried the following: $Ax = \lambda x$ $A^2x = \lambda A x $ $2Ax = \lambda A x$ But not sure where to go from here, or if this even is the right approach. Thanks to anyone who helps.
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Eigenvector centrality as a limiting case of Katz centrality

In Networks by Newman (2nd ed.), eigenvector centrality is defined as the vector $x$ that solves $$ A x = \kappa x, $$ where $\kappa$ is the principal (largest, most positive) eigenvalue of the ...
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Eigenvalues of a matrix with negative entries

I am currently working on a problem involving the spectrum of a matrix which has negative entries. More specifically, let $A$ be a matrix with entries negative entries $A_{ij}$. What conditions do we ...
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Eigenvalues for all 1's matrix doubt.

Let $J_{n \times n}$ be the all $1$'s matrix. Then find the eigenvalues of $J$. I searched around the net, was able to find the following proof: Proof. Clearly, $J$ has rank $1$. Hence there is one ...
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Matrix exponent eigenvalues

I was thinking about this for quite some time and didn't find it nowhere. So I hope someone can help me. let $e:\frak{gl}$$(n,R)$ $\to Gl(n,R)$ be the exponential application. I know that if $v$ is ...
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how to find Green function for boundary value problem

I know there's pretty generic algorithm, but I am stuck a bit. The initial problem is: $$y'' - y = f(x) \quad y'(0) = 0 \quad y(\pi) = 0$$ so I do: $$\lambda^2-1 =0 \quad \lambda_{1,2} \pm 1 $$ ...
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1answer
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How to plot phase plane of the ODE system?

I have the system (took very basic example on purpose, to understand the idea): $$\begin{cases} \dot{x} = x \\ \dot{y} = 2x -y \end{cases}$$ so I have plot phase plane. what have been done so far: $...
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1answer
28 views

T and T*T are simultaneously diagonalisable

I suspect the following might be true but I can't prove it. Suppose $T \in \text{End}(V)$ for some finite-dimensional complex inner product space $V$, such that $T^*T = TT^*$ (i.e. $T$ is normal). ...
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finding eigenvalues of $AB$ when $A$ an $B$ commute. [on hold]

Show that if $A$ and $B$ are commuting $n*n$ real matrices, then each eigenvalue of $AB$ is a product of some eigenvalue of $A$ with some eigenvalue of $B$. I know the fact that since $AB=BA$ , $A$ ...
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Eigenvalue of A and ALU

I have this matrix given and should calculate the eigenvalues: $A =\left[ \begin{array}{rrrr} -1 & 0 & 4\\ 0 & 2 & 0\\ -1 & 4 & 9\\ \end{array}\right] $ eigenvalues: $λ=2,\:λ=...
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2answers
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Eigenvalues of $T(v) = (a\times v)\times b$

Fix two elements $ a,b$ of $\mathbb{R}^3$ with $ a\cdot b\neq 0$. Let $T : \mathbb{R}^3\to \mathbb{R}^3$ be linear transformation given by $$T(v) = (a\times v)\times b$$ where $\times$ is crossproduct....
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Eigenvalues of the real part of a Hermitian matrix [duplicate]

I'm stuck trying to approach the following problem. Let $A$ be a Hermitian matrix and $B=\mbox{Re}(A)$, the real part of $A$. Show that $$\max\limits_{\mu\in\sigma(B)}\mu\leq\max\limits_{\...