1
vote
0answers
13 views

Leading eigenvalues of large sparse unsymmetric matrix

I have a matrix R which is sparse and all eigenvalues are -ve with a zero eigenvalue. Size of R is more than 1 million X 1 million. But I need to calculate only few large (by value not by magnitude) ...
0
votes
1answer
31 views

Normal Matrix Having all real eigen values is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
0
votes
2answers
41 views

How do you calculate this third eigenvector in this 3x3 matrix?

Scroll down to the bottom if you don't want to read how I arrived at my original two answers. My question is how are all the online calculators I check coming up with this third eigenvector (1, 1, ...
0
votes
0answers
30 views

About lemma $\rho(A) \leq \|A^k\|^{1/k}$

In the Spectral radius wikipedia article in section Matrices there is a lemma, what states that: Lemma. Let $A \in \mathbb{C}^{n \times n}$ be a complex-valued matrix, $\rho(A)$ its spectral ...
1
vote
1answer
59 views

Minimum eigenvalue of product of two matrices

Abstract description: Let $\mathbf{A}$ and $\mathbf{B}$ be two $n \times n$ real matrices. Let $\sigma( \mathbf{A B} )$ denote the spectrum of $\mathbf{A B}$. Assume that (A1) $\mathbf{A}$ is ...
0
votes
0answers
17 views

Which matrix norm gives the minimal variation of eigenvalues?

This is a follow-up of this question. The original question is intentionally as general as possible, because I was interested in the most general possible answer. I am now trying to understand its ...
1
vote
1answer
42 views

Is there a theorem about eigenvalues of sum of matrices?

Let's suppose I know $\lambda_i$ to be an eigenvalue of the real negative definite square matrix $A$ of size $n$. $\lambda_i$ can either be real or complex (and $\lambda_{i+1}$ will then be its ...
0
votes
3answers
36 views

Finding the eigenvectors and the diagonal of a singular 2x2 matrix

i am trying to find the eigenvectors of a 2x2 singular matrix, A = [0 , 1 ; 0 , -3]. My problem is that i can't. I know the answer is, Q = [1 , 1 ; 0 , -3] (by using Matlab), but i don't understand ...
1
vote
2answers
44 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
2
votes
1answer
30 views

A characterization of a certain family of matrices in terms of another matrix.

Consider a real matrix $A$ of dimension $n \times n$. Assume $k \leq n$ is given. I am looking for ways to describe the following set of matrices in terms of properties of $A$. $\mathcal{S}(A) = \{B ...
1
vote
2answers
86 views
+50

Lower and upper bound for the largest eigenvalue

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
2
votes
1answer
36 views

Perron–Frobenius theorem

What is exactly the Perron–Frobenius theorem? In different books papers I read different statments, and I don't know what is the truth. In wikipedia there are also a lot of statements under this ...
9
votes
1answer
121 views
+50

Theorem about positive matrices

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
1
vote
1answer
65 views

Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
5
votes
0answers
40 views

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 ...
1
vote
0answers
72 views
+50

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 ...
1
vote
1answer
46 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?
0
votes
0answers
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 ...
0
votes
0answers
8 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 $w_i \neq z_j$ ...
11
votes
3answers
290 views

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 ...
0
votes
2answers
64 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 \\ ...
1
vote
0answers
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 ...
0
votes
1answer
24 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 ...
1
vote
1answer
30 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 ...
0
votes
1answer
26 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?
0
votes
0answers
21 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 ...
1
vote
3answers
33 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?
0
votes
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 ...
1
vote
1answer
79 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 ...
0
votes
0answers
21 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 ...
1
vote
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 ...
2
votes
0answers
35 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 ...
0
votes
0answers
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 $n-1$ ...
0
votes
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 ...
0
votes
1answer
17 views

Algebraic multiplicity and similarity between rows

I know that if two rows in a square matrix are identical, one eigenvalue will have multiplicity of at least 2. I was wondering if two rows are very similar in their elements but not identical it would ...
1
vote
0answers
32 views

What is (are) the condition(s) for sum of a non-singular matrix and its transpose to be non-singular

Let a real (square) matrix $\mathbf A$ is Hurwitz (i.e., all the eigenvalues of $\mathbf A$ have negative real parts). And let $\mathbf P$ be a real symmetric positive definite matrix (i.e., $\mathbf ...
1
vote
0answers
15 views

Correlation among rows in a matrix: eigenvalues tell us anything about it?

I have a square matrix in which each rows sums to <1 and all elements are 0<=a(i,j)<=1. I am interested in measure the correlation between elements of any pair of given rows. Would the ...
0
votes
0answers
22 views

Non positive definite covariance matrix

Please help understanding which of the following would lead to a non positive definite covariance matrix and, most importantly, why? A. Changing all the correlations to be unity B. Changing all the ...
2
votes
1answer
81 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
1
vote
2answers
105 views

How do you quickly find the eigenvalues of this matrix?

I have a final exam tomorrow, am sure a 3x3 eigen value problem like the one below is there. But I find it very hard to find eigen values without zeros in the matrix Show me how you do it quickly so ...
1
vote
3answers
27 views

Matrix commutation of $\boldsymbol{M}$ and $[\boldsymbol{I} - \boldsymbol{M}]^{-1}$

We consider a matrix $\boldsymbol{M}$. We suppose it is diagonalizable, with eigenvalues $\lambda_{i}$. We always assume that $\forall i \, , \,\lambda_{i} \neq 1$. As a consequence, the matrix $[ ...
3
votes
1answer
115 views

Dominant eigenvector by looking at rows of matrix raised to a power

I'm not strong in linear algebra. I encountered this thing and being curious I want to know a bit more about it. I'm playing with 3x3 real valued matrices in some graphics application, I'm developing. ...
1
vote
2answers
70 views

When is this matrix positive semi-definite?

I have a symmetric $n \times n$ matrix as follows. I want to find the eigenvalues of this Hessian matrix to state that it is not Positive Semi-Definite (i.e. some eigenvalues are negative while the ...
0
votes
2answers
33 views

Finding an Eigenvector of 3x3 matrix

I have a question $\lambda=4$ find an Eigenvector of a given 3x3 matrix. $ A = \left[ {\begin{array}{cc} 1 & 2 & 1 \\ 6 & 1& 0 \\ -1 & -2 & -1 \end{array} } \right] $ I know ...
1
vote
0answers
53 views

Generalization of trace norm identity

Given a $2\times 2$ complex matrix $M$, the sum of its singular values (i.e. the trace norm) can be written as: $$\mathrm{Tr}\,|M|=\sqrt{\mathrm{Tr}(M^\dagger M)+2|\mathrm{Det}(M)|}$$ Is anyone aware ...
1
vote
3answers
51 views

Diagonalizability of a certain $4\times4$ matrix

Question $\bf 3.$ Determine if the following matrix is diagonalizable. (explain your answer) $$A=\pmatrix{ 1 & 4 & -2 & 3 \\ 3 & -3 & 0 & 4 \\ 1 & 1 & 1 ...
0
votes
0answers
19 views

$ || \lambda(A) - \lambda(B) ||_p \prec_k || \lambda(A -B) ||_p$?

Given two Hermitian matrices $\mathbf{A}$ and $\mathbf{B}$ and eigenvalue function $\lambda(\cdot)$ which returns eigenvalues of a matrix in non-increasing order. I found the following is true from ...
0
votes
1answer
56 views

Matrix Spectral Radius and Induced Matrix Norms

Let $A$ be a matrix, and $\rho(A)$ be its spectral radius, $\|A\|_p$ be an norm induced from vector $p$-norm. (1) When $\rho(A)=\|A\|_2$ or $\rho(A)=\|A\|_1$, does $\|A\|_1=\|A\|_2$? (2) If the ...
0
votes
0answers
47 views

Infinite Dimensional Vector Space Equivalence of Positive Matrix

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...
1
vote
1answer
31 views

Can the matrix product $PA$ be skew-symmetric with $P=P^T>0$ and $A$ Hurwitz?

Let a real (square) matrix $\mathbf A$ is Hurwitz (i.e., all the eigenvalues of $\mathbf A$ have negative real parts). And let $\mathbf P$ is a real symmetric positive definite matrix. What will be ...