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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What is known about the eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are the elements of the ...
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20 views

Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$ E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK)) $$ $$ K = 0:dK:\pi $$ Where K is a ...
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1answer
31 views

Under what conditions are the eigenvalues of a matrix finite?

Suppose we have a square matrix $A$. Under what conditions on $A$ ensure that all eigenvalues of $A$ are finite?
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19 views

Positive definite [on hold]

I need a graphical representation of positive definite from the eigen values of the matrix which can be expressed from the second degree homogenous equation.
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28 views

Positive definite matrix.

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?
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2answers
50 views

Compute an upper bound on generalized eigenvalues (by using the coefficients)

Consider the generalized, symmetric eigenvalueproblem: \begin{equation} A x = \lambda B x, \end{equation} with $A, B$ symmetric and $B$ being positive definite. For some computations, i was trying ...
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13 views

Relationship between eigen-vector and adjacency matrix nodes

My question is short and simple. I am wondering the following: lets say I have a adjacency matrix of a graph lets say NxN and λ stands for the highest eigen-valueand u for the correspondant ...
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0answers
7 views

Convex combination of polynomials with roots on the unit circle and companion matrix

Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ ...
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1answer
24 views

Eigenvalues and eigen vectors

Is it possible to have a matrix for which eigen vectors won't change by changing the eigen values? Please help me! I am seraching for the answer to this question.
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25 views

Show there exists a unique solution to $-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$

Let $\lambda\in (-1,1)$. Show that for every $f\in C[0,1]$ there exists a unique solution $u\in C[0,1]$ to $$-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$$ With $u(0)=u'(1)=0$. My work thus far: ...
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3answers
105 views

Eigen values of AB and BA

let A be a linear transformation from $R^n$ to $R^m$, and B be a linear transformation from $R^m$ to $R^n$, it's easy to show that AB and BA has same eigen-value(except $0$). But my question is how ...
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16 views

Diagonalization of sparse block matrix

I have a real symmetric matrix, \begin{equation} \left( \begin{array}{ccc} 0 & M & M' \\ M ^T & 0 & 0 \\ M ^{ \prime T} & 0 & 0 \end{array} \right) \end{equation} ...
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42 views

“Sandwich theorem” for eigenvalues of symmetric matrices

I am looking for a reference for the following result for symmetric matrices Let $A\in\mathbb R^{n\times n}$ be symmetric with eigenvalues $\lambda_n \leq\ldots\leq\lambda_1,\, M\subset \lbrace ...
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12 views

Spectrum of matrix with single scaled row

Let $M$ be a real symmetric positive-definite matrix and $D_a$ the diagonal matrix $$D_a = \left[\begin{array}{ccccc}a & & & &\\& 1 & & &\\& & 1 & ...
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32 views

Eigenvalues of a certain $3\times 3$ matrix [on hold]

could you help me to find the eigenvalues of the matrix P? Thanks in advance. Assume that three column vectors ${\bf x}_1$, ${\bf x}_2$, and ${\bf x}_3$ of dimension $N+1$ and with unitary Euclidean ...
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80 views

Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy $

Consider the operator $$ Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy $$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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+50

Eigenvalue test faster than $O\left(n^3\right)$?

Given a real $n\times n$ matrix $A$, one can find the eigenvalues in $O\left(n^3\right)$ by using say, the $QR$ algorithm. Now, what if we guess an eigenvalue $\lambda_0$, and we want to know if it's ...
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2answers
59 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
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3answers
23 views

Rank of a diagonalizable matrix?

What can be said about the rank of a diagonalizable matrix?
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1answer
35 views

Prove that for a bounded self adjoint operator, $\langle Tx,x\rangle \geq 0$ is equivalent to $\sigma(T)\subset [0,\infty)$

Prove that for a bounded self adjoint operator, the following are equivalent: A: $\langle Tx,x\rangle \geq 0$ B: $\sigma(T)\subset [0,\infty)$ What I have said so far: Since $T$ is self adjoint, ...
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2answers
56 views

Eigenvalues of a matrix that is a product of a vector and transpose vector

Find eigenvalues, eigenvectors and rank of matrix $A$. $$\textbf{a}=\begin{bmatrix}a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{bmatrix}, \quad \textbf{b} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ \vdots \\ ...
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2answers
35 views

Diagonalizing a matrix. Which formulae is correct?

In my coursebook on linear algebra on some page I see that a diagonal matrix $D$ for a matrix $A$ that can be diagonalized ca be found as follows: $$\tag{1}D=T^TAT$$ But reading further I see that my ...
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16 views

How to get transformation matrix for Linear Discriminant Analysis from eigen values?

I am trying to implement Linear Discriminant Analysis. I have 2 questions. A)Can I directly use the matrix with eigen vectors of the product of between scatter matrix inverse and within scatter ...
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1answer
109 views

Find the Jordan normal form J for A and a Jordan basis for A.

$A=\begin{pmatrix} -3&-1&1\\ -1&-3&1\\ -2&-2&0 \end{pmatrix}.$ Question: $(i)$ Determine the characteristic equation of A, hence find the eigenvalues of A. $(ii)$ Determine ...
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1answer
40 views

Trouble understanding the diagonal matrix theorem.

The Diagonal Matrix Representation Theorem states: Suppose $A=PDP^{-1}$, where $D$ is a diagonal $nxn$ matrix. If $B$ is the basis for $R^n$ formed from the columns of $P$, then $D$ is the $B$-matrix ...
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65 views

Prove that $T^n$ is diagonalizable.

Prove or give a counterexample: If $V$ is a complex vector space and $\text{dim V} = n$ and $T \in L(V)$, then $T^n$ is diagonalizable. In order to show that $T$ is diagonalizable I need to show ...
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14 views

Is there any restriction to the sum of eigenvalues for non-negative, irreduceble and square matrices?

I'm trying to find if there is a restriction in tr(A) or eigenvalues sum for a non-negative, irreducible square matrix A. As an additional information, the row sums and the order of the matrix is ...
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1answer
36 views

What is the purpose of studying Sturm-Louville eigenvalue problem?

After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential ...
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1answer
19 views

Largest eigenvalue being negative

Can anyone please tell me what can I know about a square matrix that has only non-positive eigenvalues. In a text book I read it says that lets suppose matrix A has only non-positive eigenvalues ...
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1answer
23 views

Find a basis $B$ such that the matrix has the desired shape.

Let $p=(x-(a+bi))(x-(a-bi))$ be the characteristic polynomial of linear operator $T$ ($T$ in $\Bbb C$) and its basis of eigenvector is $A=\{u+iv,u-iv\}$. Find a basis $B$ in $\Bbb R^2$ such that ...
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24 views

Lower bounds on eigenvalues of a symmetric matrix based on the diagonals

A symmetric matrix $A$ always has real eigenvalues. If I know the elements on the diagonals, is it possible to have a lower bound on the smallest eigenvalue? How sharp would this bound be? For now I ...
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1answer
23 views

How do get eigenvalues of a matrix B if add a row/column pair of a matrix A?

I have a matrix of size N×N of the form: where and A is N-1 x N-1 matrix, a=0. I known the eigenvalues of A. Any possible for getting eigenvalues of B from eigenvalues of A?
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1answer
50 views

Eigenvalues and eigenvectors of a non-symmetric matrix which is a product of 2 symmetric matrices?

I have a non symmetric matrix $AB$ where $A$ and $B$ are symmetric matrices. How can I find the eigenvectors and eigenvalues of $AB$? In a paper( Fisher Linear Discriminant Analysis by M Welling), ...
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20 views

How to get the transformation matrix for Linear Discriminant Analysis?

I am trying to implement Linear Discriminant Analysis. Is the eigen vectors of the product of within scatter matrix and between scatter matrix inverse (Sw*Sbinverse), the transformation matrix? Could ...
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3answers
30 views

non-symmetric matrix with orthogonal eigenvectors

Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?
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2answers
37 views

How does a matrix change the magnitude of a vector?

I have the following problem: $z=Ax$, in which $z$ and $x$ are $N\times 1$ vectors and $A$ is a $N\times N$ matrix. I am interested in how the magnitude of $x$ changes after applies $A$ on it. Is ...
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42 views

Estimate eigenvectors of symmetric matrix with almost vanishing diagonal

Is there a way to approximate the eigenvectors of a symmetric matrix with almost vanishing diagonal elements, i.e. with the block matrix form, \begin{equation} M=\left( \begin{array}{cc} ...
3
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3answers
42 views

How to get the two eigen vectors for eigen =1

I have to find the eigen vectors for this matrix. \begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{pmatrix} I end up with this matrix to plug in the eigen values. ...
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1answer
48 views

Relation between eigenvectors after transforming a nonsymmetric matrix to symmetric?

I need to find eigenvectors and eigenvalues of a matrix which is product of 2 symmetric positive definite matrix(SwInverseSbProd=SwInverse*Sb). Since SwInverseSbProd is non-symmetric and calculation ...
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16 views

How to convexify (relax) this L0 eigenvalue optimization problem?

Let $C_1,\dots,C_L$ be $N\times N$ hermitian matrices. Let $d<0$ be a given negative constant. Then consider the optimization problem \begin{align} \max_{r\in \mathcal{R}^{L\times 1}} &\mid\mid ...
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1answer
72 views

Finding eigenvalues of a block matrix

I have a block matrix of size $2N \times 2N$ of the form $$B = \begin{bmatrix} A_N & C_N \\ C_N & A_N \end{bmatrix}$$ where $A_N$ and $C_N$ are both $N \times N$ matrices. Specifically, $$A_N ...
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20 views

Positive definiteness and eigenvalue inequalities of two symmetric matrices

Let $A = A^T \in \mathbb{R}^{n \times n}$ and $B = B^T \in \mathbb{R}^{n \times n}$ be any symmetric matrices. Then, I know that if $A$ and $B$ are positive definite, and $A-B$ is positive ...
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1answer
18 views

Eigenvectors and the relationship between variables in a system of equations.

I am learning about complex eigenvalues in Linear Algebra and I am confused with one problem. I have a matrix in $A-\lambda I $ form. For the eigenvalue $\lambda=3+2i$, $A-\lambda I=\begin{bmatrix} ...
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16 views

Eigenvalues of correlation matrices in the limit of infinite dimensions

Consider a continuous function $f(x,t)$ with $x\in X$ and $t\in[0,1]$, then one may define a series of functions $f_n\in\mathbb{R}^n$ defined naturally as $f_n(x)_i=f(x,i/n)$. Now compare the ...
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1answer
38 views

Need help understanding the proof: if v is a left singular vector of A then v is a unit eigenvector of $AA^{T}$

This is the proof in my textbook: What I don't understand it why " $AA^{T}u = 0u $ means that u is an eigenvector. Is this a theorem that I don't know? That if you multiply a matrix by a vector ...
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34 views

Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
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24 views

Tightest upper bound for $\sum_j g_{ij}$ of an adjacency matrix of a graph

If I have an adjacency matrix of a graph $G$ (i.e. $g_{ij}=1$ if $i$ and $j$ are connected and $g_{ij}=0$ if not. $g_{ii}=0$), is there any tighter upper bound on $\sum_{j} g_{ij}$ than just $n-1$ ...
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1answer
34 views

Eigenvalues of a Matrix Using Diagonal Entries

I just started learning about complex eigenvalues and eigenvalues and one example in the book I am using says that the matrix $A = \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$. The book then ...
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2answers
17 views

Verifying eigenvalues

How would you check whether eigenvalues $\lambda_1=8$, $\lambda_2=3$, $\lambda_3=-1$ belong to a matrix? $$ \begin{matrix} 7 & 1 & 1\\ 3 & 1 & 2 \\ 1 ...
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Similar eigenvalues imply in similar rows? [closed]

My question is simple: If a matrix has rows that are similar in their elements this implies that all eigenvalues of the matrix are also similar to each other? Similarity here is defined as: B= ...