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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If an $n\times n$ matrix $A$ is diagonalizable and has only one eigenvalue $\lambda$ with multiplicity $n$, then $A = \lambda I$. True or False?

Like the title says, "If an $n\times n$ matrix $A$ is diagonalizable and has only one eigenvalue $\lambda$ with multiplicity $n$, then $A = \lambda I$. True or False?" My gut is telling me that this ...
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1answer
17 views

How to solve the eigenvalues of a complex matrix of very high condition number?

WHAT I FACE: I'm dealing with a complex matrix of very high condition number and I have to solve the eigenvalue and eigenfunction of it. But in Matlab, I got the problem that the results are not ...
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1answer
15 views

Finding the eigenvectors of a matrix that has one eigenvalue of multiplicity three

This is a simple question, which hopefully has a quick answer. I have a given matrix A, such that \begin{equation} A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 1 ...
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0answers
23 views

Show that is $\upsilon$ is an eigenvector of the matrices A and AB

(assume invertibility) Show that is $\upsilon$ is an eigenvector of the matrices A and AB with corresponding eigenvalues $\lambda \neq 0$, $\mu$ respectively, then $\upsilon$ is also a corresponding ...
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1answer
18 views

How many eigenvectors for one eigenvalue?

Consider this matrix: $\begin{bmatrix}2-\lambda & -1 & 2\\-1&2-\lambda&2\\2&2&-1-\lambda\end{bmatrix}$ and the eigenvalues $-3$,$3$ and $3$. The corresponding eigenvectors are ...
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1answer
13 views

Unique eigenvalue of maximal absolute value?

Let $A$ be an $n\times n$ matrix with $a_{ii}=0$ for all $i$, and $a_{ij}\in\{0,1\}$ for all $i\neq j$, and $a_{ij}=0\leftrightarrow a_{ji}=1$ for all $i\neq j$. Is it necessary that $A$ as a unique, ...
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1answer
30 views

Proving that an $n\times n$ matrix is positive definite iff the eigenvalues of that matrix plus its transpose are positive

I am trying to prove that an $n\times n$ matrix $A$ is positive definite iff the eigenvalues of $(A + A^T)$ are positive. So far I have: Let $x$ be an eigenvector of $(A + A^T)$ and let $\lambda$ be ...
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15 views

Eigenfuctions and Eigenvalues [on hold]

What are the eigenvalues and eignefunctions of $y''(x)+ky(x)=0$ $y'(0)=0$ $y(1)=0$
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6 views

Trace minimization-Revised

The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known. When $H$ is known, the solution is given by the eigenvectors ...
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1answer
29 views

Find the trace of the matrix $A$

Let $A$ be a $n \times n$ nonsingular complex matrix whose all eigenvalues are real.Further assume that $A$ satisfy that trace($A^2$)=trace($A^3$)=trace($A^4$).What is the trace of the matrix $A$ ? ...
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1answer
30 views

Bound for eigenvalues of some special matrix

Let $Tridiagonal(a, c, b)= \begin{vmatrix} c & b & 0 & \ldots & 0 \\ a & c & b & \ldots & 0 \\ 0 & a & c & \ldots & 0 \\ \vdots & \vdots & ...
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Find a basis and the dimension of the eigenspaces of the matrix

Find a basis and the dimension of the eigenspaces of the matrix $$ \left( \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 3 & 0 \\ 2 & 0 & 1 \end{array} \right) $$ given that the ...
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2answers
28 views

Find the eigenvectors of $ A = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix} $.

Find the eigenvectors of $$ A = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix}. $$ I know you can solve $ \det(A - \lambda I) = 0 $ to find the eigenvalues of $ A $, but I keep getting no free ...
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1answer
16 views

Eigenvalues of a Hermitian matrix and a Herminitian form

Need some help and hints on how to prove this one: Let $F=\mathbb{R}$ or $\mathbb{C}$, and $_FV=M_{n,1}(F)$. Let $A \in M_n(F)$ be Hermitian (i.e $A^* = \bar{A}^T=A$) and $f(x,y)=x^*Ay$, for all $x,y ...
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1answer
35 views

Find Eigen values of given matrix with nonfactorable polynomial

I'm having trouble finding the Eigen values for this matrix: $$ A =\begin{pmatrix} 0&1&-2 \\ 1&3&0 \\ -2&0&5 \end{pmatrix} $$ I did $A - \lambda I $ and ended up with this ...
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2answers
49 views

Understanding a proof conceptually

Let's assume that $V$ and $W$ are vector spaces over a field $\mathbb{K}$, $\lambda\in\mathbb{K}$, $\lambda\neq0$. $S: V\rightarrow W$ and $T: W\rightarrow V$ are linear maps. Prove, that $\lambda$ ...
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1answer
24 views

Eigenvalue of a linear map (proof)

Let's assume that $V$ and $W$ are vector spaces over a field $\mathbb{K}$, $\lambda\in\mathbb{K}$, $\lambda\neq0$. $S: V\rightarrow W$ and $T: W\rightarrow V$ are linear maps. Prove, that $\lambda$ ...
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0answers
24 views

Eigenvalues of antisymetric matrix

Let $A \in \mathbb{R}^{n \times n}$ be a antisymetric matrix (that is $A^T = -A$. Then $A$ han a purely imaginary eigenvalues. How to prove it? I wanted to give that as a exercise, but I am not able ...
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1answer
28 views

$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$ for all positive definite $A\in\mathbb{R}^{n\times n}$

Let $A\in\mathbb{R}^{n\times n}$ be positive definite and $v\in\mathbb{R}^n$. Let $\left\|\cdot\right\|_2$ be the Euclidean norm. Can we prove $$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$$ for ...
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2answers
44 views

Matrix Differential Equations

I am working on a practice problem with the following equation: $$ \frac{d^3 x}{dt^3} + (k + 1)\frac{d^2x}{dt^2} + (k+1)\frac{dx}{dt} + kx = 0 $$ I understand the first part which is to convert to a ...
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1answer
27 views

What does does [H1,H2] = 0 mean if each H is a hamiltonian of a quantum system

What does does [H1,H2] = 0 mean if each H is a hamiltonian of a quantum system I'm trying to get through a research paper on theoretical quantum biology and I just want to make sure I'm interpreting ...
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36 views

Exercise 2.36 in 'Vector Calculus, Linear Algebra and Differential Forms' (Hubbard)

Let $A$ be an n by n diagonal matrix with diagonal entries $\lambda_1$ to $\lambda_n$, and suppose that one of the diagonal entries, say $\lambda_k$, satisfies $inf_{k\neq j}|\lambda_k - \lambda_j| ...
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1answer
24 views

Convergence of QR algorithm to upper triangular matrix

Sorry for asking really silly question. I guess the answer will be very simple. The question I am doing is: Does QR method always converge to a upper triangular matrix? I think the answer is not. ...
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20 views

Bounding cosine of angle between vectors

Let $M$ be a symmetric, positive definite matrix such that $0\lt c_1 \le \lambda_{min}(M)\le\lambda_{max}(M)\le c_2$. I am trying to show that $\dfrac{v^TMv}{||Mv||||v||}\gt 0$ for $v\ne 0$ I ...
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0answers
12 views

Reduction of matrix $A$ to $B$ to find eigenvalues by Power method [duplicate]

How to reduce matrix $A$ to $B$ such that it has all eigenvalues and eigenvectors of $A$ but the dominant eigenvalue (eigenvalue with largest magnitude) is replace by $0$ ? I am using Power method to ...
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29 views

Name for this possible mathematical structure? [on hold]

I'm thinking of a latent variable as an nth (each representing a variable) dimensional object - it can either be for a correlation matrix, or for a factor structure in factor analysis - that's ...
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2answers
53 views

How to reconstruct a symmetric matrix given the eigenvalues and eigenvectors.

I am trying to reconstruct a symmetric 3 x 3 matrix from just its eigenvalues and eigenvectors. I think the solution involves orthogonalizing two of the eigenvectors using the Gram-Schmidt ...
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1answer
15 views

finding corresponding eigen vector to eigen value for linear system

I have a question where I am trying to find the general solution of a linearised system, which I have linearised. I am just having difficulty obtaining the correct corresponding eigenvectors to my ...
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71 views

Real matrix without real eigenvalues commutes with some matrix of square $-I$.

Let $A\in M_n(\mathbb R)$ be such that, the minimal polynomial of $A$, has not any real root. Prove that there exist some $B\in M_n(\mathbb R)$ which: $B^2=-I_n$ and $AB=BA$. Suppose that ...
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1answer
62 views

Symmetric matrix eigenvalues

Let $A$ be an $n\times n$ matrix, with $A_{ij}=i+j$. Find the eigenvalues of $A$. A student that I tutored asked me this question, and beyond working out that there are 2 nonzero eigenvalues ...
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Behavior of components of a self adjoint linear transformation split over an orthogonal sum.

Let $V$ be a finite dimensional real vector space with an inner product $<\cdot,\cdot>$. Suppose $V=A\oplus B$ and where $A$ and $B$ are orthogonal subspaces. Let $T:V\rightarrow V$ be a self ...
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What is meant by eigen spaces are non-orthogonal?

$M$ is a square matrix $M$ ( matrix representation of a linear operator $L$ acting on a hilbert space $H$ , $L: H \to H$ ) with eigen values $\lambda_i$ and corresponding eigen spaces $V_i$. I know ...
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0answers
17 views

Origin of the word “Characterstic values” of a Matrix or a Linear Operator

I found the synonyms of the mathematical term "characteristic values" of a square matrix/ linear operator. One of them is "eigen value", which itself (that I learnt) derived from a German word ...
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4 views

Eigenvalue spectrum of backward Fokker-Planck operator

I encountered a paper in physics, in which the author states that an operator of the following form (backward Fokker-Planck) $\Lambda = P(x)\frac{d}{dx}+Q\frac{d^2}{dx^2}$ has an eigenvalue 0 and ...
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Numerical Stability of inverse rayleigh quotient iteration

For the inverse iteration we solve (A-uI)w = v. So if u is close to eigen value the A-uI is poorly ill conditioned. But why this apprant pitfall in inverse iteration causes no trouble. It would be ...
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Prove if $A$ is a symmetric matrix with real entries, then the eigenvalues of $A$ are real.

Given a matrix $$A=\begin{bmatrix}a & b \\ b& c \end{bmatrix} $$ then let $\lambda = p+qi$ be a complex eigenvalue of $A$. I used the characteristic equation to get a factorizacion of ...
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How to find the other eigenvalues of a matrix when all rowsums are equal (where the rowsum provide the first eigenvalue)

If you have a situation where every row-sum is equal in a matrix A, this sum equals one of the eigenvalues of the matrix. Using this fact, is there any easy procedure/shortcut for finding the rest of ...
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1answer
64 views

Eigen value and Regular Graph (not Strongly Regular graph).

$A,B$ are 2 adjacency matrices of $d$ Regular graphs(not Strongly Regular graphs). I would like to know- 1.Results/ information related to Eigen values of A,B. There is a formula for ...
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26 views

Eigenvalues and eigenvectors for earthquake modeling

My instructor explicitly stated that, because we are asked to find eigenvalues and eigenvectors of a $7\times 7$ matrix, MATLAB would be easiest to use. The equation $(1)$ is intended to resemble ...
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1answer
27 views

Newtons Law of Cooling Differential Equations

We have two differential equations, $$\begin{cases} {dT\over dt} = -\alpha(T-B)\\ {dB\over dt} = -\beta(B-T)\end{cases}$$ If $T(0) = 7$ and $B(0) =3$, determine the equilibrium temperature of the ...
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1answer
32 views

Inhomogeneous eigenvalue problem, the shooting method and constraints

In trying to solve a problem occurring in QM calculations I've encountered the following pickle, with which I hope you could help me. I am trying to solve an inhomogeneous eigenvalue differential ...
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1answer
16 views

Proof if $A$ is normal then it is nondefective

What is the proof that if $A$ ($m\times m$ Matrix) is normal i.e $(AA^{\ast} = A^{\ast}A)$ then $A$ is non defective i.e (for each eigenvalue of $A$, its algebraic multiplicity is equal to the ...
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26 views

How to compute determinant (or eigenvalues) of this matrix?

Let us have the $n \times n$ circulant matrix given by \begin{equation} C(c_0,c_1,\cdots, c_{n-1}) =\begin{bmatrix} c_0 & c_1 & c_2 &\cdots & c_{n-1}\\ c_{n-1} & c_0 & c_1 ...
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1answer
65 views

what does it by raising a matrix to the power of $1/2$?

I came across the following which I did not understand at all. Let $A$ be a positive semi-definite. If $A(I-B)$ is positive definite, then the eigenvalues of $$A^{1/2}(I-B)A^{-1/2} = I ...
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4answers
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number of **distinct** real roots of $f(x)=0$?

Let $a$ be a non zero real number. Define $$f(x) = \begin{vmatrix} x & a & a & a\\ a & x & a & a\\ a & a & x & a\\ a & a & ...
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40 views

Eigen value of principal submatrix.

I was studying "interlacing property" and trying to find out the below fact- $A$ is an adjacency matrix of a $r$ regular graph $G$. $u,v \in G $;$u,v$ are not similar vertices. $B$ is the ...
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1answer
16 views

Solve Ax=0 using Single Value Decomposition

Trying to solve Ax=o when $A=\begin{bmatrix}2&1&-1\\1&2&1\\ \end{bmatrix}$ using single value decomposition. I have the s,v,u and was thinking that x was as simple as $x=s*s^t$ but ...
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1answer
35 views

Eigenvalues and eigenvectors in $\mathbb{R}^\mathbb{N}$?

$\mathbb{R}^\mathbb{N}$ is the vector space of all real sequences $x=(x_n)_{n\in\mathbb{N}}$. The operations are defined in this manner: $(x_n)+(y_n):=(x_n+y_n)$ and $\lambda * (x_n):=(\lambda x_n)$. ...
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1answer
12 views

Invariant subspace (Proof)

How do I prove, that the eigenspaces of $T^n$ are invariant in regard to $T$, assuming T is an endomorphism in a real vector space V $(T: V\rightarrow V)$? That's how I started: Let $E_\lambda$ be ...
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2answers
9 views

Condition for largest eigenvalue being strictly larger than largest variance

Under which condition is the largest eigenvalue of a positive semi-definite matrix strictly larger than the largest of the matrix's diagonal entries?