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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Find a and b such that the matrix is diagonalizable

Find a and b such that the matrix $$ \left( \begin{array}{ccc} 1 & a \\ 0 & b \\ \end{array} \right) $$ is diagonalizable. I know that $$ D = S^{-1} A S $$ where S is a matrix made of the ...
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26 views

Find value of k for distinct eigenvalues

Consider the matrix $$ A = \left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ k & 3 & 0 \end{array} \right) $$ where k is an arbitrary constant. For which values of k does A ...
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16 views

How to count algebraic multiplicities to show $\nexists$ an eigenbasis for $A$?

If $A=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix},f_A(\lambda)=(1-\lambda)^3 \,\text{and } E_1=\text{ker ...
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1answer
15 views

Invertible operators converging to a noninvertible operator in a finite dimensions: Eigenvalue converge to 0?

I feel like this should be an obvious property, but I want to make sure of it before I use it as the key part of a larger proof: If we have two finite dimensional vector spaces $E,F$ of the same ...
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11 views

eigenvector perturbation

In the proof of Theorem 1 of (http://ai.stanford.edu/~ang/papers/ijcai01-linkanalysis.pdf), the authors cite a theorem from Steward and Sun (Theorem V.2.8) which states that if $S$ is symmetric and ...
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27 views

Eigen space of $T_{A}$ where $T_{A}(v)=Av$

Let $T_{A}:\mathbb{C}^3\rightarrow \mathbb{C}^3$ ,$T_{A}(x,y,z)=A\begin{pmatrix}x\\ y\\ z\end{pmatrix}$ and $A=\begin{pmatrix} 1 & 5 & 0\\ 0 & 1 & 0\\ 0&0&3 \end{pmatrix}$. I ...
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1answer
26 views

Connection between algebraic multiplicity and dimension of generalized eigenspace

Assume $V$ to be a finite dimensional vector space. Define the algebraic multiplicity $am(\lambda)$of an eigenvalue $\lambda$ of a linear operator $T:V\to V$ as the maximum index of the factor ...
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2answers
61 views

Largest eigenvalues of AA' equals to A'A

I need help with proving that for any real matrix,the largest eigenvalue of AA' equals to the largest eigenvalue of A'A
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1answer
39 views

Eigenvalue of $B=uv^\text{T}+wz^\text{T}$

We have $u,v,w,z \in R^\text{n}$, how can we express the eigenvectors and eigenvalues of $B=uv^\text{T}+wz^\text{T}$ by analyzing over $u,v,w$ and $z$?
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2answers
44 views

Prove that the eigenvectors of this matrix are a basis in $\mathbb{R}^n$

Let $A \in \mathbb{R}^{n \times n}$ and $w \in \mathbb{R}^n$. Suppose that, $w_i>0$ and $a_{i,j} = w_i / w_j$ for all $i,j=1,\dots,n$. Note that from the construction comes that ...
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1answer
29 views

Intersection of Eigenvectors and Multivariable Calculus

This isn't really a problem but more of a reference/example question: do eigenvalues and eigenvectors ever show up in multivariable calculus? The two seem very unrelated to me. Specific examples would ...
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14 views

How to convert principal components of a $2\times2$ covariance matrix into principal components of a correlation matrix

All, I am wondering if there is any way to mathematically express the change in direction of the principal components from the $2\times2$ covariance matrix to the correlation matrix. In other words, ...
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1answer
17 views

Prove that the columns of the similarity matrix of a diagonalization are the eigenvectors

I'm interested in eigendecomposition of a matrix. It is clear for me, that you can eigendecompose a matrix if and only if it is diagonalizable. But I don't know how to prove, that in the similarity ...
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1answer
13 views

eigenvalues and eigenvectors of a vector twhen a multiple of identity matrix is added to the matrix?

will the eigenvalues and eigenvectors of a vector A change when a multiple of the identity matrix is added to it. for instancce (A + 3I)
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2answers
21 views

Incremental algorithm for matrix eigenvalues

I try to solve the following problem: Given a stream of symmetric matrices $A_0, A_1, ...,A_n$ such that $A_i$ is different from $A_{i-1}$ only in one place, I want to compute the eigenvalues of ...
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2answers
17 views

Similar matrices C and D: how to derive the relation $\mathbf{x} = S^{-1} \mathbf{y}$ when $C = S^{-1}DS$

D (with corresponding eigenvector $\mathbf{x}$) and C (with corresponding $\mathbf{y}$) are similar matrices, which means they have the same eigenvalues. So the relation $C = S^{-1}DS$ holds. So we ...
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2answers
32 views

Eigenvalues (or lack thereof) of $A$ for $A^2 = -I$

I'm just starting with Eigenvalues and Eigenvectors and it all seemed to be going fine until this question stumped me: Let A be a $2\times 2$ matrix for which $A^2=-I$. Prove that A has no real ...
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1answer
12 views

Unequal numbers of eigenvalues and eigenvectors in SVD?

In singular value decomposition (SVD), $X=USV^T$, if $X$ is $N\times D$, then $U$ is $N\times N$, $S$ is $N\times D$, and $V$ is $D\times D$. Let's assume $N<D$. Every column vector in $V$ is an ...
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0answers
10 views

Roots of unity: Bounds on Eigenvalues of circulant matrix

Can you tell me a bound for $$\left|\sum_{j=0}^{k^n-1}c_j e^{2\pi i j \frac{m}{k^n}}\right|, \quad m \in \{0, \dotsc, k^n-1\}$$ the absolute values of the eigenvalues of a circulant matrix with ...
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2answers
27 views

Eigenvectors and eigenvalues from $X^TX$ to $XX^T$?

I have a $N\times D$ matrix, where $N<D$. I wish to compute the eigenvalues and eigenvectors of $X^TX$, which is a $D\times D$ matrix. To speedup the MATLAB computations, I want to compute instead ...
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28 views

What is meant by an eigenvalue of 2 matrices?

In looking for a way to compare covariance matrices, I came across a paper that formulates a metric using what appears to be a joint eigenvalue. I'm not familiar with this idea. Thus we propose ...
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15 views

Eigenvalues special block matrix

Let us consider the block matrix $$M=\left( \begin{matrix} 0 & Id \\ A & B \end{matrix} \right) $$ where $Id$ denotes the identity matrix $N\times N$ and $0,A,B$ have the same dimensions. Do ...
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1answer
23 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I don't have a large mathematical background, but I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular ...
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11 views

If A1 and A2 are irreducible row stochastic matrices, can we prove that (I-A1)(A2-I) is stable?

$A_1\in \mathcal{R}^{n\times n}$ and $A_2\in \mathcal{R}^{n\times n}$ are row stochastic (row-sum-1) but not necessarily symmetric, and $I\in \mathcal{R}^{n\times n}$ is the identity matrix. Can we ...
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0answers
17 views

Convergence of Arnoldi method

I would like to compute the largest real eigenvalue of a matrix in the following form: $$\begin{bmatrix} 0 & I_n \\ P & Q \end{bmatrix},$$ where $I_n$ is the $n \times n$ identity matrix, $P$, ...
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0answers
26 views

eigenvalues inequality finite differences

I have $x,y\in[0,1]^2$, $a\in[0,A]$ $t\in[0,T]$ and the mesh points $x_j = j \, \Delta x, j=0,\ldots,J$; $y_l = l \, \Delta y, l=0,\ldots,L$; $a_k = k\Delta a, k=0,...,K$ and $t_n = nh, n=0,\ldots,N$ ...
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2answers
62 views

If $BA$ has $-1$ as an eigenvalue, then so does $AB$?

I was just encountered with a rather tough problem as follows: Suppose $A,B\in M_n(\mathbb R)$, prove: $$\det(I_n+AB)\ne0\Rightarrow\det(I_n+BA)\ne0$$ Although at this moment I am still at a ...
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31 views

T,S: V->V, proove that TS and ST have the same eigenvalues

hey I was trying to proove this proposition by deviding to cases and this is what i've got so far: let's assume without loss of generality that T is invertible: ST = [T][S] TS = [S][T] ...
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1answer
27 views

how to prove if [T]b is diagonal then there is a scalar “a” such that T(v)=av

hey i was trying to prove the next proposition: given T:V->V for every Basis B, if the matrix [T]B is diagonal, then there is a scalar "a" for every v in V such that T(v)=av this is what i managed ...
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2answers
41 views

How to prove that the minimum eigenvalue of the difference of two specific matrices is negative

Let us define two matrices: Matrix $D$: diagonal matrix with positive entries Matrix $A=s\cdot s^{T}$, where $s$ is a vector with norm one and non-zero entries. Therefore, $A$ is symmetric with ...
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1answer
36 views

Mapping of the eigenvector of eigenvalue 1 to a different matrix

Let $M \in (0,1)^{n\times n}$ be an irreducible and primitive column stochastic matrix. Then for the Perron theorem, $\exists x^* : Mx^* = x^*$. We want to build a matrix $K \in \mathbb{R}^{n\times ...
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0answers
18 views

Diagonal decomposition, square root and eigenvector / eigenvalue of a matrix

I have encountered a problem of finding eigenvector and eigen value of a matrix of type $$ A = \dfrac{1}{2} \begin{pmatrix} 4&1&-2\\ -4&1&6\\ 2&0&-2 \end{pmatrix} $$ Also I ...
3
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2answers
67 views

Eigenvalues and Spectrum

In algebra, I learned that if $\lambda$ is an eigenvalue of a linear operator $T$, I can have \begin{equation} Tx = \lambda x \tag{1} \end{equation} for some $x\neq 0$, which is equivalent to $\lambda ...
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17 views

A calculation for an open ball in $\mathbb{R}^N$ and function space.

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by ...
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2answers
42 views

Proof of a real eigenvalue

Let $A$ be a $2\times2$ matrix $A=\begin{pmatrix}a&b \\ c&d\end{pmatrix}$. I found the characteristic polynomial which is $T^2-(a+b)T+ad-bc$. It can be written as ...
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27 views

How to find eigen vector for an eigen value in generalized eigen value problem

I have a generalised eigen value problem of the form $A$x = λ$B$x. I have computed the eigen value (say λ1) I am interested in using Eigen library(C++). However, because the library does not support ...
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30 views

Finding the eigenvalues and eigenvectors with each eigenvalue, solving the general solution with initial conditions.

Consider the system $x'_1 = x_1 + 2x_2$ and $x'_2 = 3x_1 + 2x_2$ If we write in matrix from as $X' = AX$, then a) $X =$ b) $X' =$ c) $A =$ d) Find the eigenvalues of A. e) Find eigenvectors ...
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1answer
91 views
+50

How to differentiate between $(\lambda_{0}-\lambda)^{k} \,\text{and } g(\lambda) \,\text{in } f_{A}(\lambda)$?

By definition, $\lambda_{0}$ has algebraic multiplicity $k$ if $\lambda_{0}$ is a root of $f_{A}(\lambda)=(\lambda_{0}-\lambda)^{k}g(\lambda)$. What am I missing from this? ...
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35 views

Inequality with eigenvalues

Let matrix $ X $ is Hermitian and denote $ \lambda_1(X) \ge \lambda_2(X) \ge \ldots \ge \lambda_n(X) $ eigenvalues of matrix $ X $. Prove that $ \lambda_i(A + B) \le \lambda_i(A) + \lambda_1(B) $ I ...
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3answers
62 views

Find the eigenvalues of $A$. $A^2 = 1$ and $A\ne\pm1$

$A \in \mathbb{R}^{n\times n}$, with $A^2 = 1$ and $A\ne\pm1$ Show that the only eigenvalues of $A$ are $1$ and $-1$.
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1answer
29 views

Linear Algebra-invariant subspaces

Suppose $V$ is a real vector space and $T\in \mathcal L (V)$ has no (real) eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.
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42 views

Gradient of the eigenline that corresponds to eigenvalue

The matrix A is given by Matrix $A = \begin{pmatrix}3&-23\\-13&-2\end{pmatrix}$ The matrix A has two eigenvalues h and k, where h > k. To 2 decimal places, what is the gradient of the ...
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1answer
20 views

Principal Component analysis by eigenvalue decomposition.

I do know how to perform PCA by using SVD but I am unaware about how to use eigenvalue decomposition of X(transpose)*X matrix. I found a paper online which explains the approach to perform PCA by ...
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52 views

How to find the eigenline equation

Matrix $A_{2\times 2} = \begin{pmatrix}24&10\\12&19\end{pmatrix}$ Eigenvalues are: $\frac{43+\sqrt{505}}{2}$ OR $\frac{43-\sqrt{505}}{2}$ With those, 1st eigenvalue, ...
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2answers
41 views

Finding complex eigenvalues

For the matrix \begin{pmatrix}1/2 & 1 & 3/4\\2/3 & 0 & 0\\0 & 1/3 & 0\end{pmatrix} Find the eigenvalues and corresponding eigenvectors. I did this with an online calculator and ...
2
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1answer
43 views

Find orthogonal Q given eigenvalue and eigenvector?

Given some upper Hessenberg matrix $H \in R^{n \text{x} n}$, i know how to find an orthogonal matrix which is a product of Givens rotations such that $P^THP$ is also upper Hessenberg, but I'm not sure ...
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1answer
31 views

Proof that an $n \times n$ matrix is positive definite iff all of its eigenvalues are positive

I am trying to prove that an $n \times n$ matrix is positive definite iff all of its eigenvalues are positive. I know that if $\lambda$ is an eigenvalue then: $Ax = \lambda x$ for eigenvalues ...
0
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1answer
29 views

Relationship between eigenvectors of two matrices

Suppose I have matrix $A \in R^{2n \text{x} 2n} $ given by $X^{-1} diag(W - iY, W + iY) X$ and matrix $B \in C^{n \text{x} n}$ and $B = W + iY$. Let $v$ be an eigenvector of $A$. How can I relate ...
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30 views

Proving that eigenvalues are positive iff $det(A_k)> 0$ for all $k = 1, …, n$ for a real symmetric matrix $A$

I am trying to prove that eigenvalues of $A$ are positive iff $det(A_k)> 0$ for all $k = 1, ..., n$ for a real symmetric matrix $A$ where $A_k$ is the $k \times k$ matrix obtained by deleting the ...
0
votes
1answer
59 views

Show convergence of Power method

Given a symmetric positive definite matrix $A_0 \in R^{n \text{x} n}$ with Cholesky decomposition $A_0 = LL^T$. How can I show that $A_k$ converges to $diag(\lambda_1, ..., \lambda_n)$ where $A_k$ is ...