1
vote
1answer
21 views

Finding similar matricies

I'm trying to find a matrix N similar to the scalar matrix M = $ \begin{pmatrix} a & 0 \\ 0 & a \\ \end{pmatrix} $ Such that $M = ANA^{-1}$. I have no idea ...
0
votes
1answer
23 views

Know eigenvalues, get $Q$ of $A=QLQ'$

$A=\begin{bmatrix} 1 & -2 & 2\\ -2 & -2 & 4\\ 2 & 4 & -2 \end{bmatrix}$ I have calculated that the eigenvalues $\lambda=2,2,-7$. When $\lambda=2$, the eigenvector is ...
0
votes
1answer
25 views

Why does Givens rotation avoid iteration and Jacobi rotation doesn't in case of reducing a symmetric matrix to tridiagonal?

I am currently implementing symmetric matrix reduction to tridiagonal. I read that Givens rotation avoids iteration when it is used for reducing a matrix to tridiagonal whereas Jacobi rotation is ...
1
vote
0answers
29 views

Eigenvalues with constraints?

Note: This is a short version of About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms) For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists ...
0
votes
1answer
23 views

Linear Algebra - Give an example for $3x3$ matrix for these eigenvalues

I'm having trouble with this problem : Give an example for matrix $A$ with these eigenvalues $\lambda_1-1,\lambda_2=1,\lambda_3=0$ while : $$v_1=(0,1,1)$$ $$v_2=(1,-1,1)$$ $$v_3=(0,1,-1)$$ ...
2
votes
1answer
27 views

Get normalised eigenvectors

I am given the matrix: $\begin{pmatrix} a & b \\ b & -a \end{pmatrix}$ and I already calculated the eigenvalues $\lambda = \pm \sqrt{a^2+b^2}$. Now, I want to get the normalised ...
2
votes
3answers
62 views

Roots of a cubic equation with coefficients based on unknown values $a$, $b$ and $c$.

I want to find the eigenvalues of the following matrix: $$ \left( \begin{array}{ccc} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right) $$ So, I found the characteristic ...
1
vote
3answers
47 views

Prove that a matrix with a given characteristic polynomial is diagonalizable

Matrix $A$ is defined over real number. Characteristic polynomial : $p(x)=(x+3)^2(x-1)(x-5)$ It also known that : $$\text{rank}(A+2I)+\text{rank}(A+3I)+\text{rank}(A-5I)=9$$ prove $A$ ...
-1
votes
1answer
31 views

Linear Algebra - Prove trival solution eigenvalue

A is an $2\times2$ matrix with $\operatorname{trace}=1$, and $\det A=-6$. Prove that $(2A+5I)x=0$ has only trival solution. I need to show that $(-A-\frac{5}{2}I)x=0$ Therefore I need to show that ...
1
vote
0answers
17 views

median eigenvalue

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
6
votes
1answer
58 views

What is the quickest way to find the characteristic polynomial of this matrix?

Let $e_k$ be the $k$-th vector of the canonical base of $\mathbb R^n$ and let $$B = [e_2 \mid e_3 \mid \dots \mid e_n \mid e_1]$$ What it the quickest way to show that the charachteristic polynomial ...
1
vote
0answers
40 views

Diagonalization of Hermitian matrix

I would like to perform diagonalization of a Hermitian matrix $A$ and I know the steps but at the end I am not getting diagonal matrix with eigenvalues on the main diagonal, can anyone help me why? ...
0
votes
2answers
34 views

Two Matrices with Negative Eigenvalues of Each Other?

I have two matrices, $A$ and $B$. I was (perhaps naively) expecting them to be more-or-less similar ("more-or-less" because this is in a numerical setting), but instead of having exactly the same ...
0
votes
0answers
12 views

Eigenvalues of Hankel matrices

Let $\mathbf{A}$ be a $4-$ dimensional symmetric matrix with real entries, whose elements are given as \begin{equation} \mathbf{A} = \left( \begin{array}{cccc} a & b & c & d \\ b & c ...
3
votes
2answers
32 views

Eigenvalues of 3D rotation matrix

I'm having some trouble calculating the eigenvalues for this rotation matrix, I know that you subtract a $\lambda$ from each diagonal term and take the determinant and solve the equation for $\lambda$ ...
1
vote
3answers
38 views

Eigenvectors of $\left( \begin{array}{ccc} 0 & -b \\ a & 0 \end{array} \right)$

This is similar to my previous question in that I when I form a system of simultaneous equations and solve them all the terms cancel and I don't get any information on the eigenvectors. The matrix in ...
1
vote
3answers
65 views

Eigenvectors of $\left( \begin{array}{ccc} a & 0 \\ 0 & -b \end{array} \right)$

I calculated the eigenvalues of the following matrix to be $a$ and $-b$. $J = \left( \begin{array}{ccc} a & 0 \\ 0 & -b \end{array} \right)$ But when I use the formula $(J - \lambda I)v = 0$ ...
0
votes
2answers
30 views

Show that the inverse of a strictly diagonally dominant matrix is monotone

I have been struggling with this problem for awhile. Given that $A$ is a strictly diagonally dominant matrix with positive diagonal entries and non-positive off-diagonal entries, show that $A$ is ...
0
votes
1answer
26 views

When does a square matrix have an eigen-decomposition? When is a matrix defective? [duplicate]

Some square matrices, like $ \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)$, don't have a complete set of eigenvectors. By complete I mean that the eigenvectors span the entire ...
1
vote
2answers
98 views

Possible eigenvalues of a matrix $AB$

Let matrices $A$, $B\in{M_2}(\mathbb{R})$, such that $A^2=B^2=I$, where $I$ is identity matrix. Why can numbers $3+2\sqrt2$ and $3-2\sqrt2$ be eigenvalues for the Matrix $AB$? Can numbers $2,1/2$ ...
4
votes
2answers
118 views

The rank and eigenvalues of the operator $T(M) = AM - MA$ on the space of matrices

This problem is from Artin Algebra Second edition, 5.2.3. Let $A$ be a $n\times n$ complex matrix. (a) Consider the linear operator $T$ defined on the space $\mathbb{C}^{n\times n}$ of all complex ...
0
votes
1answer
21 views

Eigenvalue and eigenvector of $A'A$

Suppose that $\mathbf{A}\in\mathrm{R}^{m\times m}$ is a square but not necessarily symmetric matrix whose eigenvalues and eigenvectors are $\lambda_i$ and $\mathbf{x}_i,$ $i = 1,2,\cdots,m$. Is ...
2
votes
1answer
25 views

Parallelism in Golub & van Loan's Jacobi algorithm for symmetric eigenvalue problems

In Matrix Computations by Golub and Van Loan (3rd edition, page 433) an algorithm is given for a parallel version of the classical Jacobi algorithm for solving a real symmetric eigenvalue problem. The ...
0
votes
1answer
25 views

Eigenvectors for shear matrix and diagonalizing.

Here is a shear matrix $ \begin{pmatrix} 1 && 0 \\ 2 && 1 \end{pmatrix}$. The eigenvalues are 1. $ \lambda^2 - 2 \lambda + 1 \to \lambda = 1$. So now I try to find the eigenvectors. ...
0
votes
1answer
40 views

Confusion on Eigenvalues of Matrix

I'm a TA with Advanced Algebra in school and teach the Jordan Form now. There are three questions about eigenvalues in this chapter: Given matrix $A$, $B$ and polynome $f$, consider the eigenvalues' ...
1
vote
1answer
28 views

matrix inequality related with eigenvalues

I was asked to prove the following: Let $A \in M_{n}(\mathbb{C})$, then there is an Hermitian matrix $H$ and an skew-Hermitian matrix $K$ such that $A=H+K$. If $\sigma(A) = \{\lambda_1, \lambda_2, ...
7
votes
2answers
115 views

Sum of squares of maximal minors of a rectangular matrix with orthonormal rows

A matrix $A$ has $m$ rows and $n$ columns, such that $m \leq n$. We know that each row of $A$ has norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
1
vote
0answers
30 views

Find the characteristic polynomial and the eigenvalues $\lambda_1, \lambda_2,\dots,\lambda_n$ of the matrix $A$

Let $\lambda_1, \lambda_2, \dots, \lambda_n$ be eigenvalues of the matrix $A=(a_{ij})_{n\times n}$. Is it true that $|A|=\lambda_a\lambda_2\dots\lambda_n$, and $tr(A)=\displaystyle\sum_i^n\lambda_i$. ...
0
votes
1answer
21 views

Find the eigenvalues of T and an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix

Find the eigenvalues of T and an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix. $V = P_1(R)$ and $T(ax+b) = (-6a + 2b) x + (-6a + b)$ First i gave the canonical basis of ...
0
votes
0answers
27 views

If I have a real symmetric matrix and one of the diagonal elements are zero, does that say anything about the eigenvalues?

In the two dimensional case, if I have a matrix $\left[\begin{array}{cc} 0 & a\\ a & b \end{array}\right]$ or $\left[\begin{array}{cc} b & a\\ a & 0 \end{array}\right]$ then I have a ...
0
votes
1answer
23 views

Why does this expression for the minimum eigenvalue work?

I read that we can have, for a column vector $d \in R^N$, \begin{eqnarray} min_d = \frac{d' A d}{d'd} = \lambda_{min} \end{eqnarray} , where $\lambda_{min}$ is the smallest eigenvalue of the ...
2
votes
3answers
56 views

To prove that $A$ has a one-dimensional eigenspace , where $A \in SO(3)$ , $A \ne I$

Let $A\ne I$ be a $3\times3$ real orthogonal matrix with determinant $1$ , then how to prove that $A$ has a one-dimensional eigenspace ?
0
votes
0answers
19 views

proving bounds by bounding eigenvalues?

I have an expression that takes the form:$\mu_n= \displaystyle (I_d+\sum_{i=1}^n f_i f_i^T)^{-1}(\displaystyle\sum_{i=1}^n f_i)$ where $f_i$'s are d dimensional vectors with bounded $L^2$ norm. $I_d$ ...
1
vote
0answers
18 views

Eigensystem of direct sum of matrices with diagonal elements of different order of magnitude

I have got a problem with matrices like, for example: $\left( \begin{array}{cccccc} 1 & 1 & 2 & 1 & 1 & 2 \\ 1 & 1 & 1 & 1 & 3 & 1 \\ 2 & 1 & 1 & ...
0
votes
1answer
39 views

Eigenvalues & eigenvectors of a matrix

I have a couple of questions regarding eigenvalues and eigenvectors. Let $A=\begin{pmatrix}4 & 2 \\ 5 & 1\end{pmatrix}$, $\mathbf{u}=\begin{pmatrix}2\\-5\end{pmatrix},\mathbf{v}=-2\mathbf{u}$ ...
2
votes
3answers
67 views

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, then $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, show that $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$. I have already been able to show that if $A$ is an arbitrary ...
1
vote
3answers
66 views

To prove that the sum of the roots of the characteristic polynomial of a square matrix is equal to the trace of the matrix

How do we prove that the sum of the roots of the characteristic polynomial of a square matrix is equal to the trace of the matrix ? I want a proof which does not use much computation or determinants ; ...
0
votes
1answer
29 views

Minimal polynomial of f restricted to its image

Let $f:V\to V$ be a $F$-linear map, $V$ an $n$-dimensional vector space over $F$, $\operatorname{rank} E=r$, $W=\operatorname{Im} f$, $\tilde f:=f|_W:W\to W$. Let $\mu$ be the minimal polynomial of ...
2
votes
1answer
44 views

Generalized eigenvalues of overdetermined systems

I have a system of equations that can be written as ${(\bf{A}} + \lambda{\bf{B}}){\bf{x}} = 0$ Where ${\bf{A}}$ and ${\bf{B}}$ are $n \times m$, integer matrices. I know that there are several ...
0
votes
0answers
31 views

Eigenvector and 2D Rotation

I have a problem where i have a 2x2 matrix and need to rotate the coordinate system to make it a diagonal matrix. The solution involves calculating the eigenvector of this matrix. Considering that a ...
0
votes
1answer
18 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 ...
0
votes
0answers
14 views

Issues with connecting the SVD and Eigenvalues for block matrix

In class, we have talked about the singular value decomposition and its connection to Eigenvalues. Specifically, for a matrix A, if the columns of a matrix contain linearly independent eigenvectors, ...
0
votes
4answers
52 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
106 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
35 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
27 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
56 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
36 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
114 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 ...