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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Calculating the Eigenvectors and Eigenvalues of this Matrix Polynomial

For the matrix $$ A=\begin{pmatrix} 1 & 1 & 2 \\ 0 & -2 & 0 \\ 0 & 2 & 3 \end{pmatrix} $$ How are the eigenvalues and eigenvectors of the following matrices calculated? ...
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Real or imaginary eigenvalues?

The question I have been lost in for a while is when will a matrix have either all real or complex eigenvalues? (Depending on dimensions of the matrix in question, complex and real eigenvalues may ...
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Cannot find eigenvectors

How can I find eigenvectors of the following matrix? $$ \begin{matrix} 4 & 0 \\ 0 & 1 \\ \end{matrix} $$ Systematic approach would be: 1. Finding eigenvalues ...
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Is it possible to diagonalize a singular matrix?

I have not seen anywhere written that it is impossible, but it seems impossible, so I want to check if I missed something. According to a theorem, an nxn matrix is diagonalizable if it has n ...
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Proof that $e^x$ is the eigenvector or the derivative operator

I remember hearing my professor talk about how $e^x$ shows up in all our differential equations because it is the eigenvector for the derivative operator. Can someone explain and prove this to me? I ...
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1answer
26 views

Finding Eigenvalue for cubic equation

I'm learning finding eigenvalues. I learned how to find simplistic eigenvalues for $3\times3$ matrix. By using below way. With this way I can only solve if I have simple determinant equation, like ...
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38 views

how to relate the eigenvalues and eigenvectors of these two matrices?

If $W, Y \in R^{n \times n}$, then how the eigenvectors and eigenvalues of these two matrices are related? $C = W +iY, B = \begin{bmatrix} W & -Y\\ Y & W\\ \end{bmatrix} $ Specifically, ...
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1answer
18 views

Projection on cone of non-negative definite matrices

Ok, so if you have a real symmetric matrix $Q$ then the projection of that matrix on the cone of symmetric non-negative definite matrices $\mathcal{C}$ can be explicitly found if we do an ...
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37 views

how this computes the eigenvalues?

I've read somewhere that the following iteration referred to as "inverse orthogonal iteration" (I don't know why?) can be used to compute the $p$ smallest eigenvalues of $A$ in absolute value. I ...
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28 views

Proof concerning eigen values

Could somebody help me into proving this theorem? if $A$ and $B^{H}$ are in $C^{m\times n}$ with $m\geq n$, then $\lambda (AB) = \lambda(BA) \cup \lbrace 0, \ldots ,0\rbrace.$ Thenks, Elnaz
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orthonormal vector properties

I have noticed a matrix property that is outlined below: I have a set of n orthonormal eigenvectors that form a basis in Rn. If these vectors are combined to form an nxn matrix where each column is ...
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49 views

How to find all $3\times3$ matrices $A$ that satisfies $A^2-3A-4I = 0$? [on hold]

How to find all $3\times3$ matrices $A$ that satisfies $A^2-3A-4I = 0$?
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51 views

For which $a$ is a matrix $A$ diagonalizable?

Say I have a matrix $A_a$ with $$A_a:= \left(\begin{array}{c} 2 & a+1 & 0 \\ -a & -3a & -a \\ a & 3a+2 & a+2 \end{array}\right)$$ I was wondering if there was an ...
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25 views

A Challenging Question. Eigenvalues of a Special Matrix.

$N\times N$ matrix $L$ can be partitioned into $p\times p$ blocks, ...
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2answers
28 views

Significance of an eigenvector being equal to a unit vector?

I was reading ahead in my math book when I came across a matrix denoted as A = $\begin{bmatrix} 1 & -1 & 0\\ 2 & -2 & 0\\ 6 & 0 & -2\\ \end{bmatrix}$. I then found the ...
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0answers
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eigenvalues of homogeneous integral equation of second kind, with singular kernel

There is a homogeneous integral equation of second kind with a singular kernel(non-symmetric). The equation has the form: $\int_{a}^{b} k(x,t)Γ(t)dt =λΓ(x).$ It's 2-norm is infinity, $||k(x,t)||_2 ...
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How to find the rules by which eig(A) = diagonal entries

Suppose A = [a b c;d e f;g h i] how to find a series of rules such that eig(A) = a, e, i such as h = 0 or f = 0? Is there any rule for this? (I know we can find this by principal submatrix, but I ...
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What does my teacher mean by 'choosing' from a vector?

I'm revising some lecture notes from a class I missed, I'm just struggling to figure out what she means at this point. What is choosing x1=0, x2=1... etc mean? Could someone explain?
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Can centrality of given network decreased on removal of higest central node

Actually I am calculating the Eigenvector Centrality of whole graph with each node centrality also.Then I remove the highest central node(high eigenvector centrality) in graph by making respective ...
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A quick way to estimate eigenvector/eigenvalue of a matrix

Is there a quick way to give a raw estimation of an eigenvector/eigenvalue of a matrix? By "quick" I mean some method which can be computed without a computer or paper and pencil...something you could ...
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3answers
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How to get An eigenvalue and eigenvectors of a matrix that contain both zero column and zero row?

Could anyone help in how to get the eigenvalue and eigenvectors of a matrix that contain both zero column and zero row like : \begin{pmatrix} -1 & 1 & 0\\ 1 & -1 & 0\\ 0 ...
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The lower bound of Cheeger Inequality as the degree goes to infinity

Consider an undirected graph $G(V,E)$ with adjacency matrix $A$ and define the graph Laplacian as \begin{equation} L = D - A \end{equation} where $D$ is a diagonal matrix such that $D(i,i) = d_i$. ...
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For two p.d. matrices $A$ and $B$, prove that $\lambda_1(AB)\leqslant \lambda_1(A) \cdot\lambda_1(B)$

If $A$ and $B$ are two nxn positive definite matrices, then show that $$\lambda_1(AB) \leqslant \lambda_1(A) \cdot \lambda_1(B),$$ where $\lambda_1(\cdot)$ denotes the largest eigenvalue.
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Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5

Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5 . Then, there exists a non-zero vector $v$ in $R^2$ such that (a) $||Av||$ > 2$||v||$; (b) $||Av||$ < 1/2$||v||$; ...
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Are eigenvalues of A all with positive real parts if and only if $x^TAx>0$?

Are eigenvalues of $A$ all with positive real parts if and only if $x^TAx>0$ for any $x$? $A$ is non symmetric. If this is true, if $B=-B^T$, then if the eigenvalue of $A$ are with positive real ...
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58 views

eigenvectors of a matrix

Good Day, I have a matrix of \begin{bmatrix} 28 & 10\\ 10 & 19 \end{bmatrix} I have found the eigenvalues... first eigenvalue (v1) : 24 + 5sqrt5 = rd off to 35.18 second eigenvalue (v2) ...
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25 views

On the eigenvalues / properties of a specific matrix.

I'm not sure how to better phrase the title of the question, because I don't know the specific name of the matrix I am after, but I want to consider matrices of the form $$ \begin{align*} ...
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19 views

A question about eigenvalues of a special block matrix

Thanks for anyone who views or answers this question! $N\times N$ matrix $L$ can be partitioned into $p\times p$ blocks, ...
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2answers
54 views

Showing that matrix admits an eigenvector?

Let A= a b c d be a 2 x 2 matrix, where a,b,c and d are real numbers. We say that A admits an eigenvector if there exists a unit vector u and a real ...
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Why do we want that the determinant of the coefficients is $0$?

Eigenvalue problem with periodic boundary conditions-complete Fourier series $$y''+\lambda y=0, 0 \leq x \leq L$$ $$(*): \begin{cases} y(0)=y(L)\\[4pt] y'(0)=y'(L) \end{cases}$$ $$$$ It's a ...
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57 views

When do eigenvectors converge?

Let $A_n$ be a sequence of self-adjoint $N\times N$ matrices that converge in the operator norm to $A$. The sequence of eigenvalues of $A_n$, denoted $\lambda_n$, converges to an eigenvalue of $A$, ...
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Hilbert-Schmidt theorem

In the Hilbert-Schmidt theorem what it means : $A e_n=\lambda_n e_n$ ? Thank you .
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General eigenspace & general eigenvector

Consider the following: 1.Suppose $k_i$ is an eigenvalue of $A$ with algebraic multiplicity $n_i$ 2.dim $V_i = m_i$ (geometric multiplicity), $V_i$ is the eigenspace corresponding to $k_i$. 3. $m_i ...
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1answer
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How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

I have the following matrix $$ \begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix} $$ First I got the eigenvalues which are $0$, $1$, $2$. I tried to get the ...
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Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
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1answer
46 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
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Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
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Finding characteristic roots and characteristic vectors

V is a two-dimensional vector space over the field of real numbers, with a basis $v_1, v_2$. Find the characteristic roots and corresponding characteristic vectors for T defined by $v_1(T) = v_1 + ...
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Any transformation in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$. [closed]

$V$ is a two-dimensional vector space over a field $\mathbb F$; prove that every element in $A(V)$, the ring of linear transformations on $V$, satisfies a polynomial of degree $2$ over $\mathbb F$.
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How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
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1answer
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Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
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44 views

Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
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What are eigenvalues of higher order finite differences matrices?

I know (at first empirically, then read somewhere) that for for second order finite differences matrices like $$\begin{pmatrix} -2&1&0&0&0\\ 1&-2&1&0&0\\ ...
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1answer
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Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
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eigen values and vectors in this question

I have the following matrix to be solved for landa. here is the matrix with my answer (sorry for the wrong equation writing format): but the correct equation from this matrix should be (when ...
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Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is ...
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1answer
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Find an $n\times n$ integer matrix with determinant $1$ and $n$ distinct positive eigenvalues

I feel pretty stupid for doing this, but here goes anyway. Earlier today I asked: Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues. As it turns out, for my problem I ...
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3-Species Population Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
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Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
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1answer
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Calculating eigenvectors where there is only 1 non zero number in matrix

So I am attempting to find the equilibrium points of a nonlinear system and I am getting the following jacobian matrix: $$ \begin{pmatrix} 1 & 0 \\ 0 & 1.5 \\ ...