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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4
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1answer
37 views

Powers of 2 matrices having identical row sums

I have been working on a research problem and have encountered a situation in which two matrices $A, B \in \mathbb{R}^{n \times n}$ are such that \begin{equation} A^k \mathbf{1} = B^k \mathbf{1} \...
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1answer
20 views

Show that $|\lambda_i(A)|<1$ iff $|\lambda_i(\beta A)|<1$ $\forall \beta: |\beta|\leq 1$

Here $\lambda_i(A)$ is the $i$-th eigenvalue of the square matrix $A$. I would like to know if these two inequalities are equivalent. I assumed they are (please correct me if I am wrong). So I tried ...
2
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1answer
36 views

Which is true about $Q$ where $Q=I+2P$

Let ${a_{1},a_{2},...a_{n}}$ and ${b_{1},b_{2},...b_{n}}$ be two bases of $\mathbb{R}^{n}.$ Let P be an $n \times n$ matrix with real entries such that $Pa_{i}=b_{i}$ for $i=1,2, ...,n.$ Suppose that ...
1
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0answers
20 views

Solving a generalized eigenvalue problem with constraints

I have the following generalized eigenvalue problem: $ \begin{pmatrix} 0 & a \\ a^T & B / \lambda_{i} \end{pmatrix} \begin{pmatrix} 1 / \lambda_{i} \\ x_{i} \end{pmatrix} =\epsilon_i \begin{...
0
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0answers
10 views

Singular values of the product of two (semi)orthogonal matrices

Let's assume we are given two (semi)orthogonal matrices $U_1$ and $U_2$ with dimension $m\times n$ such that $m>n$. The (semi)orthogonality means $$U_1^TU_1 = I_n$$ and $$U_2^TU_2 = I_n$$ but ...
1
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0answers
34 views

Condition number of a $2\times 2$ square block matrix

Is there a general rule to relate the condition number of the $2\times2$ square block matrix $ \left(\begin{array}{cc} A & B\\ C & D\\ \end{array}\right), $ where the matrices have the ...
0
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0answers
29 views

If I have n different eigenvalues prove their eigenvectors are linealy independent [duplicate]

Prove via induction that if $V$ is a vector space of finite dimension and T: $V\to V$ a linear operator with n different eigenvalues then the eigenvectors associated with them are linearly independent....
5
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1answer
26 views

Derivative of a characteristic polynomial at an eigenvalue

Let $p(\lambda)$ be the characteristic polynomial of an $n\times n$ matrix $A$. We know that the roots of $p(\lambda)$ are the eigenvalues of $A$, hence the sum of the roots of the polynomial (taking ...
1
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1answer
60 views

Spectral theorem for diagonlizable matrices

For a diagonalizable matrix $\textbf A_{n \times n}$ with spectrum $σ(\textbf A)=\{\lambda_1, \lambda_2,..., \lambda_k\}$ we have matrices $\{ \textbf G_1, \textbf G_2,..., \textbf G_k \}$ such that: ...
0
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0answers
24 views

How to estimate the product of the $k$ largest eigenvalues of a matrix

Now I have a question which let me to prove that the product of the largest $k$ singular values of a real matrix is always larger than the one of $k$ largest eigenvalues. For $k=1$, I use the ...
1
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1answer
71 views
+50

How to start an eigenvalue problem

I am stuck on this problem : This is an eigenvalue problem $$\phi''+ \lambda^2 x(x+2)^2 \phi =0\\\phi(1)=0\\ \phi(0)=0$$ I forget this kind of problems... please give me a hint or a clue ,cause I ...
0
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1answer
37 views

interpretation of eigenvalue

Let $A$ be a symmetric, positive definite matrix. An eigendecomposition of $A$ produces $A$ = $W V W^T$, where $W$ is $A$'s corresponding eigenvector matrix and $W$ is $A$'s corresponding (diagonal) ...
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0answers
10 views

Convergence of a stationary iteration method for linear systems

Recently, I obtain a linear system, $Ax = b$, where $A$ is a nonsingular, strictly diagonally dominant $M$-matrix. Then I also got a matrix splitting $A = S - T$, where $S$ is also a nonsingular, ...
0
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1answer
30 views

Eigenvalue of multiplicity k of a real symmetric matrix has exactly k linearly independent eigenvector

If A is an nxn real symmetric matrix then A is diagonalisable. In other words, If A is a symmetric nxn matrix, then there exists an orthogonal matrix $P$ such that $P_{-1}AP=P_{T}AP=D$, a diagonal ...
9
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3answers
367 views

Matrices that are not diagonal or triangular, whose eigenvalues are the diagonal elements

I want to learn about matrices whose diagonal elements are the eigenvalues... but the matrix is neither diagonal nor triangular. Is there a term for such matrices, and have they been researched?
0
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1answer
26 views

Find $P$ such that $A=P^{-1}JP$ where $A$ is the matrix of $f$ and J is the Jordan Form. $P$ non invertible?

Find the Jordan Form and a basis of Jordan for the endomorphism of $R^4$ $$f(x,y,z,t)=(x,x+y-t,-2x+y+z+2t,-x+2t)$$ After doing all the process, I find $P$ such that $A=P^{-1}JP$ where $A$ is ...
0
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0answers
75 views

Spectrum of unit hyperspheres in large dimensions

Each eigenvalue of $\mathbb{S}^n$ is $k(k+n-2)$ with multiplicity $((n^2-3n+2)+ 2(n-1)k)\binom{n+k-3}{k}$. I would like to confirm that result in using the Weyl's law (see Spectral Geometry in Non-...
3
votes
1answer
50 views

Eiegenvalue equation

I have a matrix $M = D X X^T$, where $D$ is a diagonal matrix with real entries, and $X$ is a $n \times d$ matrix. Note that $M$ is not symmetric. I want to find the vectors $\alpha$ for which: $$X^T ...
2
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1answer
58 views

Problem about linear algebra [duplicate]

Suppose we have two $n \times n$ square matrices A and B such that $AB=BA$. It is known that A, B and AB all have n distinct eigenvectors that is a basis of $\mathbb{C}^n$. Can we then show that there ...
0
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0answers
13 views

How many eigenvectors can a real, symmetric, sparse matrix have?

I'm attempting to diagonalise a sparse, real and symmetric matrix H in MATLAB, using [F, E] = eigs(H, size(H,1)). This however ...
1
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1answer
17 views

Inverse of a quasipositive matrix with negative spectral bound

A square matrix is quasipositive if all off-diagonal elements are nonnegative. The spectral bound of a square matrix is defined as $$s(A) = \max\{\Re (\lambda) : \lambda \mbox{ is an eigenvalue of } A\...
2
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1answer
34 views

Spectrum of a triangle; Beltrami operator

I would like to find the spectrum of a triangle (e.g. an equilateral) in using the usual Laplacian. I am not able to find some references on the subject. I'd try to solve the problem myself with a ...
2
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1answer
23 views

Help with computing eigenvalues

If we know that $A$ is a $3 \times 3$ symmetric matrix, and we have eigenvalue $\lambda_1 = 0$ with eigenvector $a=(0,0,1)$, eigenvalue $\lambda_2=1$ with eigenvector $b=(2,1,0)$, and $\lambda_3 < ...
0
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0answers
5 views

Can the transformation matrix in eigenvalue problems be considered as the identity matrix multiplied by a scalar?

I tried doing some eigenvalue problems and tried to convert the given matrix into the form of an identity matrix multiplied by a scalar and thus obtaining the values of that scalar but the values ...
0
votes
1answer
22 views

Eigenvalues and eigenvectors for a quasi-circulant matrix

Related to this question: here There is a well-known closed form expression for eigenvalues and eigenvectors of a circulant matrix. For example, see Wikipedia https://en.wikipedia.org/wiki/...
0
votes
1answer
31 views

Given $TT^* = 4T - 3I,$ Prove $T$ is positive definite and find all eigenvalues of $T$.

Let $T$ be a linear transformation in a complex finite dimensional vector space equipped with a positive definite inner product. Suppose that $TT^* = 4T - 3I,$ where $I$ is the identity and $T^*$ is ...
0
votes
1answer
31 views

First Order Difference Equations - Using Eigenvectors/Values

I was reading some notes and there was the following section: Start with a given vector $\vec{u}_0$. We can create a sequence of vectors in which each new vector is $A$ times the previous vector: $$\...
0
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0answers
49 views

Find the eigenvalues the block matrix $M=\begin{bmatrix}A+2D & A \\ A & D \end{bmatrix}$

Let $A$ be any square matrix with eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ and $D$ is a diagonal matrix with entries $d_1,d_2,\cdots,d_n$, then how can one find the eigenvalues of the ...
0
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0answers
23 views

Relationship between geometric multiplicity, algebraic multiplicity and left and right eigenvectors of a matrix

The following statement is from the book Matrix Analysis by Horn and Johnson. An eigenvalue λ with geometric multiplicity 1 can have algebraic multiplicity 2 or more, but this can happen only if ...
0
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1answer
21 views

Confused with the reexpression of a Hamiltonian in eigenbasis

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...
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2answers
70 views

How to determine which of the following matrices are similar?

If we have the following three matrices: $$ A=\begin{bmatrix} 7 &1 \\ -5 &3 \end{bmatrix},\;\; B=\begin{bmatrix} 5 &-1 \\ 1 &5 \end{bmatrix},\;\; C=\begin{bmatrix} 5 &1 \\ 1 &...
0
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2answers
42 views

Repeated use of Woodbury formula

I want to calculate the $x$ dependency of $\left(I + A \Lambda (x) A^{T}+B\Omega(x)B^{T}\right)^{-1}$ explicitly, where $I$ is a $n\times n$ matrix. Here $\Lambda (x) $ and $\Omega(x)$ are diagonal $...
-1
votes
0answers
57 views

$2 \times 2$ block matrix related

let $A$ be any matrix of order $n$, $J$ is matrix of order $n$ whose all entries are $1$, and $I$ is an identity matrix of order $n$, then how to find eigenvalues of following block matrix? $$M=\...
1
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1answer
41 views

Eigenvalues of linear transformation $f(a_n)=a_{n+1}-a_n$

The linear transformation $f:\mathbb{R^N}\to \mathbb{R^N}$ with $\mathbb{R^N}$ being the $\mathbb{R}$-Vectorspace of all sequences $(a_n)_{n\ge1}$ is defined as $$f \big( (a_n)_{n \ge 1} \big) = ( a_{...
1
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1answer
48 views

Eigenvalues and eigenvectors of $I \otimes A \ + \ B^T \otimes I$ (used in Sylvester's equation)

Let $A$ and $B$ be $n \times n$ square matrices, with resp. eigenpairs $(\lambda_i,U_i)$ and $(\mu_j,V_j)$. Let $I_n$ be the order $n$ identity matrix. I have seen a result that says that the $n^2$...
0
votes
0answers
28 views

First steps in derivation of matrices spectrum

I was trying to go through a paper about 'The eigenvalue spectrum of a large symmetric random matrix' by Edwards and Jones (1976) and I found myself stuck at the very first step of a derivation. I ...
0
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0answers
13 views

Meaning of notation $L_{subscript}$ in ridge detection.

In the wikipedia article on ridge detection, it says "let $L_{pp}$ and $L_{qq}$ denote the eigenvalues of the Hessian matrix \begin{pmatrix} L_{xx} & L_{xy} \\ L_{xy} & L_{yy} \end{pmatrix}...
2
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2answers
42 views

Diagonalizable by orthonormal matrix

Given the matrix $$A = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$ Explain why $A$ can be diagonalized by an orthonormal matrix and find an ...
1
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0answers
22 views

Boundary value problems: eigenvalue and eigenfunction

I'm having trouble in understanding eigenvalues and eigenfunctions in BvP the problem is: $y''$ + $\lambda$$y$ = $0$ $y(0)=0$ $y(2\pi)$ = $0$. Make characteristic polynomial $r^2 + \...
2
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1answer
19 views

Show that $q(T)(x)=\sum_{n=1}^\infty q(\lambda_n) \langle x,e_n\rangle e_n$ coincide with $q(T)=\sum_{k=0}^n a_kT^k$

Let $Tx=\sum_{n=1}^\infty \lambda_n \langle x,e_n\rangle e_n$ be bounded where $\{\lambda_n\}_n$ are the complex eigenvalues and $\{e_n\}_n$ are an orthonormal basis of the separable space $H$. For ...
0
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0answers
13 views

Determine eigenvalue distribution support

I am working on a project regarding random matrix spectra and I need some help with the following: let us assume we are looking at some particular family of NxN random matrices in the limit of N -> ...
0
votes
0answers
15 views

Given A $m\times n$, $\{u_1,…,u_n\}$ ON basis of $R^n$, prove eigenvalues aren't negative [duplicate]

Given $A$ $(m\times n)$, $\{u_1,...,u_n\}$ ON(orthonormal) basis of $R^n$ which are eigenvectors of $A^TA$ with $\lambda_1 , ... , \lambda_n$ eigenvalues accordingly. Prove: Eigenvalues are not ...
0
votes
0answers
54 views

Find the Eigenvalues of Petersen Graph

Petersen graph is k-regular graph on $n$ vertices and $m$ edges. We can find eigenvalue of $k-regular$ graph by characteristic polynomials of $G$ (denote $\chi_G (x)$) and $L(G)$ (denote $\chi_L (x)$)...
0
votes
2answers
273 views

Min-Max Principle with matrices - Understanding $\lambda_2$

Related to the question Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ - Explanations, I am trying to do the same with $$A= \begin{bmatrix} 2 & 0 & 0\\ 0 &...
2
votes
0answers
37 views

explicit self adjoint operator which has no diagonalization

Let a linear operator $T : H \to H$ be diagonalizable if $H$ has an orthonormal basis composed of eigenvectors of $H$ Give an example of an explicit self adjoint operator which has no diagonalization ...
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1answer
19 views

Find the value of n , using eigenvector

I am unable to think how shall I proceed. I have to find value of n given a 2×2 matrix and an eigenvector. Can somebody help me out.
1
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1answer
43 views

Operator norm and eigenvalue inequality

Can I say that $\|A\| < s$ where $A \in \mathbb{R}^{3 \times 3}$ is a symmetric, positive definite matrix and $s$ is the maximum eigenvalue of $A$. Here the norm used is operator norm.
2
votes
1answer
42 views

What is $\mbox{Tr}^2(A)-\mbox{Tr}(A^2)$ in terms of the eigenvalues of $A$?

I am looking for a way to relate the terms of the characteristic polynomial of a $3 \times 3$ matrix to its eigenvalues. The definition I start with (taken from Wolfram MathWorld) is $\\P_{3}(A)=x^{...
1
vote
1answer
38 views

Find SVD of $A$

How do I find the singular values? They somehow show that $\lambda_1 = 27, \lambda_2 = 6, \lambda_3 = 0$. I still can't see how they found them with the equations I made in my solution.
1
vote
1answer
74 views

Show map is norm-preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...