0
votes
1answer
16 views

Effect of the nature of noise on the spectrum of a random matrix

Consider the following two equations $X = M + \eta_1$ $Y = M + \eta_2$ where, $X\in\mathrm{R}^{n\times n}$, ia a real random matrix with mean $M\in\mathrm{R}^{n\times n}$. $\eta_1$ is Gaussian ...
-1
votes
0answers
24 views

matrices and eigen values [NBHM-2014] [on hold]

In each of the following cases, describe the smallest subset of $\Bbb{C}$ which contains all the eigenvalues of every member of the set $S$. a. $S = \{A ∈ M_n(\Bbb{C}) | A = BB^*, B \in ...
0
votes
4answers
44 views

Finding the diagonalizing matrix.

Find a nonsingular matrix $C$ such that $C^{-1}AC$ is a diagonal matrix. $$ A=\begin{pmatrix} 1 & 0 \\ 1 & 3 \\ \end{pmatrix} $$ I have found the eigenvalues to be 1 ...
3
votes
1answer
91 views

How prove this matrix $B^{-1}-A^{-1}$ is positive-semidefinite matrix,if $A-B$ is positive matrix

Question: Let $A,B$ be positive $n\times n$ matrices, and assume that $A-B$ is also a positive definite matrix. Show that $$B^{-1}-A^{-1}$$ is a positive definite matrix too. My idea: ...
1
vote
2answers
32 views

Eigenvalues of the product of two matrices

Let $A$ and $B$ be $m \times n$ and $n \times m$ real matrices. I was asked to prove that if $\lambda$ is a nonzero eigenvalue of the $m \times m$ matrix $AB$ then it is also an eigenvalue of the $n ...
0
votes
0answers
24 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
1
vote
0answers
52 views

Question about condition number $k$ of a matrix over a finite field

If $\lambda_{max}$, and $\lambda_{min}$ denote the maximum and minimum values of the eigenvalues of a normal square matrix repectively- are there any explicit bounds to the eigenvalues of such a ...
-1
votes
1answer
23 views

properties of largest eignvalue of product of two matrices

I'm searching for the proof of this lemma it's about largest eignvalue of product of two matrices. one of them is positive definete and the other one is symmetric. B is symmetric matrix, A is Positive ...
0
votes
1answer
25 views

Cholesky Decomposition and Orthogonalization

I recently came across a methodology for orthogonalizing variables that are collinear, that uses Cholesky Decomposition, but I am not entirely grasping the intuition of it. Let' assume we have three ...
1
vote
2answers
110 views

How do I show that $1$ is not an eigenvalue for $A$, by showing that there are no eigenvectors for $\lambda = 1$.

Consider the matrix $$A=\begin{pmatrix}-1 & 3& 3& 3\\ 3& 1& -1& 5\\ 3& -1& 7& -1\\ 3&5& -1&1\end{pmatrix}.$$ How do I show that $1$ is not an ...
1
vote
0answers
15 views

Prove one of the eigenvector entries has the smallest magnitude

Let $L\in \mathbb{R}^{n \times n}$ be the Laplacian matrix of a simple undirected graph and $D_i$ be the same size matrix with $i$th diagonal element $1$. Denote the smallest eigenvalue of $L+D_i$ as ...
0
votes
0answers
27 views

Find the limit of this matrix as its power approaches infinity

Find the matrix power, Ak, of A = (v1,v2) v1 = (p,1-p) v2 = (1-p',p') Where v1 and v2 are column vectors, and 0 <= p <= 1, 0 <= q <= 1, p /= q. ...
-1
votes
2answers
52 views

If a matrix is non diagonalizable, what other method can I use to calculate the nth power?

First off, I have this matrix A: 1 0 3 1 0 2 0 5 0 I have calculated the eigenvalues, which are ...
0
votes
1answer
32 views

How to solve matrix eigenvalue equation which has a summation.

General problem: If I have some $n \times n$ matrices $\mathsf{M}^\tau$, and column vectors (with $n$ rows) $X^\tau$ is there some mathematical tricks I can do to solve the eigenvalue equation $ ...
2
votes
2answers
53 views

Eigenvalues of a special $M \times M$ matrix

I could not obtain an explicit formula for the eigenvalues of matrix $$ \begin{pmatrix} a & b & 0 & 0 & 0 & \cdots & 0 \\ c & a & b & 0 & 0 & \cdots & 0 ...
0
votes
1answer
23 views

On matrices sharing the same smallest nonzero eigenvalue and related eigenvector

Suppose that $A$ and $B$ are square matrices with the proper size, then what kind of condition does $B$ have to satisfy such that $A$ and $AB$ share the same smallest(largest) nonzero eigenvalue and ...
1
vote
2answers
29 views

Eigenvalue and proper subespace.

I have the follow problem: Suppose that $A,B\in{\cal M}_n(\mathbb{R})$ such that $AB = BA.$ Show that if $v$ is an eigenvector of $A$ associated to the eigenvalue $\lambda$, with $Bv\neq 0$ and ...
0
votes
1answer
37 views

How do you diagonalize this matrix and find P and D such that A = PDP^-1?

1 1 4 0 -4 0 -5 -1 -8 I3 = 3x3 identity matrix λ 0 0 λI3 = 0 λ 0 0 0 λ λ-1 -1 -4 = 0 λ+4 0 5 1 ...
4
votes
1answer
81 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
1
vote
0answers
17 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
45 views

A normal matrix with real eigenvalues 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
44 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
38 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
85 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
18 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
40 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
54 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
55 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 ...
2
votes
2answers
128 views

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
42 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 ...
13
votes
3answers
332 views

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
67 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
45 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
81 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 ...
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
18 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
14 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
296 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
66 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
16 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
25 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
38 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
23 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
42 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
87 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
24 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
39 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 ...