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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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How to solve this Linear Algebra question? [closed]

Let \begin{pmatrix} 0 & 0 & 0 & a \\ -1 & 0 & 0 & b \\ 0 & -1 & 0 & c \\ 0 & 0 & -1 & d \end{pmatrix} Suppose 0(Zero) is an eigenvalue of A with ...
Prasanna lk's user avatar
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How to show a matrix DAD has distinct eigenvalues, where D is a diagonal matrix and A is a highly structured matrix

If D is a positive diagonal matrix with well-separated diagonal entries (in particular, $(1 + k) |D_{i - 1, i - 1} < D_{i, i} < (1 - k) D_{i + 1, i + 1}$, where $k$ is a constant and the ...
Stephen Jiang's user avatar
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Eigenvalues of A, B, and (A+B) [closed]

Could the following statement be correct? If all the real parts of the eigenvalues of matrices A and B are negative, then all the eigenvalues of (A+B) also have negative real parts. Alternatively, the ...
Babak Taran's user avatar
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polarization ellipse for complex eigenvalues corresponding the phase and eigenstates.

I want to draw polarization ellipses at 0.0 eV, 0.12 eV, 0.16 eV, and 0.2 eV for my transmission eigen-polarization-values plots using eigenphase data (in radians). I've attached the final result ...
Anshul Bhardwaj's user avatar
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Prove that a square rank $1$ matrix can have at most one eigenvalue different from $0$

I'm sure this can be proven in other ways, but I'm curious if the gap in the line of reasoning presented below can be filled so that the proof is valid. Proof by contradiction. Assume the opposite: ...
powerline's user avatar
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Eigenvectors of two commuting diagonalizable matrices when the eigenspaces need not have dimension one

Let $A,B$ be commuting diagonalizable $n\times n$ matrices over $\Bbb C$. Suppose that the eigenvalues of $A$'s are all distinct (so the eigenspaces have dimension one), and the same for $B$. Then any ...
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If $\lambda$ is an eigenvalue of $T$, then $|\lambda|^{2}\le\displaystyle\sum_{j=1}^{n}\sum_{k=1}^{n}|M(T)_{j,k}|^2$

This is a problem from Axler's "Linear Algebra Done Right", 4th Edition, problem 19 of section 6A: Suppose $v_1,\dots,v_n$ is a basis of $V$ and $T\in L(V)$. Prove that if $\lambda$ is an ...
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compute eigenvectors of an $n\times n$ linear transform knowing the eigenvectors of an $(n-1)\times(n-1)$ linear transform on a projected subspace

Supposed I'm trying to find the eigenvectors and eigenvalues of a square $n\times n$ matrix $A_{n}$. Further suppose that I've applied some algorithm and identified one eigenvalue and eigenvector ...
Jim Newton's user avatar
2 votes
1 answer
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Eigenvalues of a banded matrix

I want to know the eigenvalues of given matrix $\bf{K}$, which is defined as $${\bf K}=\begin{bmatrix} \bf{A} & \bf{X} & \bf{O} & \bf{O}\\ \bf{X} & \bf{B} & \bf{X} & \bf{O}\\ \...
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What are the properties of the incidence matrcies of undirected graphs

Here is the definition of incidence matrix I find on Wikipedia https://www.wikiwand.com/en/Incidence_matrix Suppose we have a graph $G$ with $N$ nodes and $e$ edges (we only consider undirected graphs ...
LeoB's user avatar
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help me understand the Spectral theorem for the laplacian

Let $u:Ω→R$ be the solution of: $∆u=λu$ and $u=0$ on $∂Ω$ Let $S=$ The spectrum of $∆ =$ all the values of $λ$ for which there is a solution. If I understand correctly if $Ω$ is bounded, we have the ...
Alucard-o Ming's user avatar
1 vote
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l2-norm of matrix products

I have a matrix $Q\in\mathbb{R}^{m\times n}(m>n)$, where the columns of Q are unit and mutually orthogonal. $W\in \mathbb{R}^{m\times m}$ is a diagonal matrix with diagonal elements 0 or 1, and $WQ$...
kai deng's user avatar
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Spectral abscissa of a real matrix achieved at a real eigenvector

Consider a real matrix $A\in\mathbb{R}^{N\times N}$ with eigenvalues $\left\{\lambda_i\in\mathbb{C}\right\}_{i\in[N]}$ and the spectral abscissa $\alpha(A)=\max_{\lambda_i(A)}Re(\lambda_i)$ achieved ...
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$u,v$ are unit vectors. What is the value of max $|u^TAv|$

Suppose $u,v\in \mathbb{R}^n,A\in\mathbb{R}^{n\times n}$, $u,v$ are unit vectors. The goal is to get the value $\max_{u,v}$ $|u^TAv|$. If $A$ is symmetric, then it can be diagonized. $|u^TAv|= \langle ...
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Eigenvalue Spectrum of a non-Hermitian Matrix by a Hermitian Matrix Perturbation

Let $H_{eff}$ be a $n$ dimensional matrix defined by the eigenvalue spectrum $\Lambda$: $$\Lambda(H)_n=\Lambda(H_n+H_{eff}),$$ Where $H$ is a infinite dimensional matrix, $\Lambda(H)_n$ are its lowest ...
do.t.rian's user avatar
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2 answers
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Quadratic form of a real symmetric matrix is bounded

If $\lambda_1>\lambda_2>...>\lambda_r$ are the different eiegenvues of a real symmetric matrix $A\in M_{n\times n}(\mathbb{R})$. $1.$ Show that the quadratic form associated to $A$ satisfies ...
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Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices

Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2 ,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume ...
Ruben Verresen's user avatar
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Can components of an eigenvector be deduced using linear algebra methods?

In the following equation $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1\end{bmatrix} \psi = E \psi$$ by ...
James's user avatar
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Let $A \in K^{n, n}$, $B \in K^{m,m}$ and $C \in K^{n,m}$ with $rank(C) = m$ and $AC = CB$. Every eigenvalue of B is also an eigenvalue of A.

Let K be a field, $n \ge m$ and $A \in K^{n, n}$, $B \in K^{m,m}$ and $C \in K^{n,m}$ with $rank(C) = m$ and $AC = CB$. Show, that every eigenvalue of B is also an eigenvalue of A. My anwser so far: ...
furnio's user avatar
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Off -diagonal perturbations of a positive semi-difinite matrix

Does the following statement hold? For a positive semi-definite(PSD) matrix X, if $rank(X) \ge 2 $, there exists a nonzero off-diagonal matrix D that both X+D, X-D are also PSD. If it doesn't work, ...
Junsukim's user avatar
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Why do numerical eigenvector computations turn out different compared to symbolic formulas?

The following $A \vec x=\lambda \vec x$ should have 4 eigenvector solutions of the form Setting $c=1, m=1, p_x=1, p_y=1, p_z=1$, the eigenvalues should be $$\lambda = \pm 2$$ while the corresponding ...
James's user avatar
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Why solve SVD sign ambiguity (indeterminacy) problem choose inner product the singular vector and the individual data vectors?

onsider a matrix A = USV' A = [1 2;3 4;5 6] the matrix U is [-0.23 0.97;-0.97 -0.23;0.06 -0.01] choose the first column of A as the individual data vector a1 = (1,3,5) the inner product of u1 is u1 * ...
dians's user avatar
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Eigenvalues of superoperators and their Choi matrices

It is well known that $\Phi$ is a completely-positive and trace-preserving (CPTP) map if and only if the corresponding Choi matrix $C_\Phi:=\sum_{i,j} E_{i,j}\otimes \Phi(E_{i,j})$ is positive semi-...
Thinkpad's user avatar
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How to determine the sign of sensitivity for the eigenvalues with respect to the elements of a square matrix?

For a square matrix $\mathbf{A}$, its right eigenvector $\mathbf{u}$ satisfying: $$\mathbf{A}\mathbf{u} = \lambda\mathbf{u}$$ , where $\lambda$ is one of the eigenvalues of $\mathbf{A}$. Besides, the ...
Kuonji's user avatar
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1 answer
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What algorithm can compute the eigenvectors when eigenvalues are already known?

Suppose eigenvalues for a matrix has been found previously. There is a human-friendly way to solve for the eigenvectors, for example, How can the above "human algorithm" be programmed on a ...
James's user avatar
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$A(z) \in \mathbb{C}[z]^{n \times n}$ - Exploit matrix structure to find analytical eigenvalues for $n>4$

I have a complex polynomial matrix $A(z) \in \mathbb{C}[z]^{n \times n}$. Generally such a matrix does not allow to analytically calculate the eigenvalue polynomials for $n>4$. However, does anyone ...
Bastian's user avatar
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Real part of an eigenvalue lies in numerical range

I‘m struggling to prove this Lemma and will be glad to get hints: If $A\in\mathbb{R}^{n*n}$, $\lambda$ is an eigenvalue of $A$, then $Re(\lambda)\in\{\frac{x^TAx}{x^Tx}, x\in\mathbb{R}^n$\ $\{0\}\}$ ...
veirab's user avatar
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Understanding the proof of Perron–Frobenius theorem

I reviewed the proof for Perron–Frobenius' theorem as stated in this article. In the proof, they define Q to be a positive orthant ( $Q:= \{ x \in \mathbb{R}^n: x \geq 0, x \neq 0 \}$) For an ...
malaiyur-mambattiyan's user avatar
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Why is there a focus on positive solutions of elliptic eigenvalue problems?

I am looking at the theory of principal eigenvalue problems of elliptic operators. A simple form would be \begin{align} \label{eu_eqn} Lu(x) &= \lambda u(x) \quad x \in \Omega \\ u(x) &= 0 \...
Ozymandias's user avatar
1 vote
1 answer
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Eigenvectors of a matrix that commutes with subgroup of permutation matrices

I have a certain symmetric matrix $M_{ij}$, that is invariant under certain permutations of some of the indices $i=1,\dots,N$. More precisely, there exists a subgroup of permutations $G$, such that if ...
a06e's user avatar
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1 vote
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Eigenfunction of "curl" are orthogonal

Let Ω be open, $(C^∞ (Ω))^3$=V , $v∈V$ such that $∇×v=λv$. Define $⟨u,v⟩=∫_Ω u_1 v_1+u_2 v_2+u_3 v_3 dx$. It is easy to see that $⟨∇×u,v⟩=⟨u,∇×v⟩$. I want to prove that if $u,v$ are 2 eigenvectors of ...
Alucard-o Ming's user avatar
1 vote
0 answers
19 views

Free probability version of Poincaré Separation Theorem

Suppose $A$ is a $d \times d$ real positive semi-definite matrix, and $U$ is a $d \times n$ semi-orthogonal matrix such that $U^\top U = I_n$. Define $B = U^\top A U$. The Poincaré Separation Theorem ...
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1 answer
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eigenvalues of contraction

Assume $A - B$ is a contraction, i.e., its spectral radius is smaller than $1$. We also assume $A$ is diagonalizable. I am trying to show that if $x^TB=0$, then $x^T$ can be written as a sum of ...
Morad's user avatar
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Eigenvalue decomposition of submatrix

for two rank- 1 matrices $\boldsymbol{A}$ and $\boldsymbol{A}_{\mathrm{sub}}$ I am interested in the relation between the eigenvalue decompositions (EVD) of ... \begin{align} &\boldsymbol{A} = \...
Dennis Marx's user avatar
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For a square matrix $A$, if $\dim(ker(A - 2I) = 3$, then its characteristic polynomial is of the form ($\lambda - 2)^3$ * q ($\lambda$)

For a square matrix $A$, if $\dim(ker(A - 2I) = 3$, then its characteristic polynomial is of the form ($\lambda - 2)^3$ * q ($\lambda$) for some polynomial q. My approach: So we know that A has an ...
brodar's user avatar
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7 votes
5 answers
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The connection between determinants and eigenvalues

I'm trying to prove the following proposition: Let $A\in \operatorname{Mat}_n\left(\mathbb{R}\right)$ be a matrix with no real eigenvalues. Then $\det\left(A\right)>0$ I know how to prove it when ...
Blabla's user avatar
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1 answer
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Follow-up question regarding a constant polynomial and eigenvalues.

I have a question regarding the answer given to this question: Proof that a is an eigen value of p(T) if and only if a=p(lambda) for some eigenvalue lambda of T Note. We are not assuming $V$ is finite ...
Paul Ash's user avatar
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3 votes
4 answers
149 views

Understanding the implication in linear algebra regarding vectors

Let $V$ be a subspace of $\mathbb{R}^n$ with the usual dot product, and let $\mathbf{z}, \mathbf{w} \in V$ be fixed vectors. If for every $\mathbf{v} \in V$ it holds that $\mathbf{z} \cdot \mathbf{v} =...
brodar's user avatar
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derivation of a third-order normal component tensor of a fourth-order tensor

I have the following constitutive material fourth-order tensor $\boldsymbol{\mathcal{C}}$ \begin{equation} \boldsymbol{\mathcal{C}}= \frac{\partial \boldsymbol{P}}{\partial\boldsymbol{F}}, \quad ...
Khoder Alshaar's user avatar
2 votes
1 answer
36 views

coupled eigenvalue problem with two eigenvalues

how can I find solutions ($\omega_i$ and $\mathbf{v}$) for a system which looks like this: $$\begin{pmatrix} A_1 & A_2 \\ 0 & 0 \end{pmatrix}\mathbf{v} =\omega_1 \mathbf{v}\\ \begin{pmatrix} ...
ZaraReinm.'s user avatar
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Finding eigenvalues and eigenvectors of a operator in $L^2[0,1]$

Find the eigenvalue and eigenvector of $T: L^2[0,1] \to L^2[0,1]$ given by $$\forall x\in [0,1],\forall f\in L^2[0,1],\ (Tf)(x)=\int_0^xf(t)dt$$ I already check that $ T \in \mathcal{LC}(L^2[0,1])$, $...
Nicolas Rodriguez's user avatar
1 vote
1 answer
44 views

Linear transformations and non-eigenvectors

Suppose $X$ is an infinite dimensional vector space and $T:X \to X$ is a bijective linear transformation different from the identity: Can one find countably infinite many linear independent vectors ...
Markus's user avatar
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2 votes
1 answer
43 views

Singular value of a bidiagonal matrix?

Consider a $n\times n$ matrix: \begin{equation} X=\begin{bmatrix} a &1-a & & \\ & a &1-a & \\ & & \ddots &\\ & & & 1-a\\ & & & a \end{...
Heydude's user avatar
  • 304
3 votes
2 answers
251 views

Prove that if eigenvalues are all different then eigenvectors are linearly independent

I was trying to work out a proof of the fact that if eigenvalues are all different, then the eigenvectors are linearly independent. Is the following proof fine or does it have errors ? Given that $ \...
Tirthankar's user avatar
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1 answer
99 views

How can we prove that there exists a $0\neq v\in\Bbb C^n$ such that $Av=av, Bv=bv$.

Let $A,B$ be two $n\times n$ complex matrices, $AB=BA$. $a,b\in\Bbb C$ satisfies: if the polynomial $f(x,y)$ verifies that $f(A,B)=0$, then $f(a,b)=0$. How can we prove that there exists a $0\neq v\in\...
xldd's user avatar
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Block Matrix eigenvalues

I am working on a complicated optimization problem. I define $I\in \mathbb{N}, I > 1, n_I=\binom{I}{2}$. I try to optimize a scalar-valued function $\mathcal{O} : \mathbb{R}^n \longrightarrow \...
Goug's user avatar
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Eigenvalues of a circulant matrix

So we define a circulant matrix as follows. Let the first column be $$a^t = [a_0, a_1, a_2, a_3, \dots,a_{n-1}].$$ The other entries are given by circularly permuting the column 1. For example column ...
user880941's user avatar
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Find values which make two matrices congruent

For which values of $a\in\mathbb{R}$ are the following matrices congruent? $$A=\begin{pmatrix} 1&4-a-a^2\\ 2& -1 \end{pmatrix}$$ $$B=\begin{pmatrix} -a-1 & 3\\ 3 & -5 \end{pmatrix}$$ ...
user926356's user avatar
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1 vote
0 answers
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Numerically stable calculation of eigenvector derivatives for repeated eigenvalues

Let us consider a real symmetric matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ with an eigenvalue $\lambda$ of multiplicity $m$. Furthermore, let $\mathbf{X}=[\mathbf{x}_1, \mathbf{x}_2,\ldots, \...
TobiR's user avatar
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0 votes
1 answer
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Spectrum of operator $T:l_1\to l_1, Tx = (0,0,x_5,2x_1,0,x_1+x_3,0,0,...)$

What is the spectrum of an operator $T: l_1 \to l_1$, $x = (x_1,x_2,...,x_n,...)$, $Tx = (0,0,x_5,2x_1,0,x_1+x_3,0,0,...)$? For $\lambda \ne 0$, equation $Tx=\lambda x$ doesn't have non-zero solutions....
Taras's user avatar
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