0
votes
1answer
9 views

Example of a special kind of infinite dimensional vector space and a linear map on it

Give example of an infinite dimensional vector-space $V$ and a linear transform $T$ on $V$ such that $T \circ S=S\circ T , \forall S \in \mathscr L(V) $ , but $V$ has a non-zero vector which is not ...
0
votes
1answer
12 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
1answer
19 views

Numerically find all zeros of multivariate function

How do I find all zeros of a multivariate function , i.e. f(x1,x2,x2,...xn)=0 numerically? I don't know exact analytic form of f , but can numerically compute f at every point on its domain. ...
0
votes
0answers
23 views

matrix diagonalization without eigen decomposition, what other ways available?

I have a matrix, $A$ (it may be symmetric or asymmetric). I need to have a diagonal matrix without eigenvalue decomposition, please suggest what others ways are possible? Any new idea would be much ...
0
votes
0answers
23 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 ...
5
votes
2answers
44 views

Finding the eigenvalues of a given Markov matrix

Let $$A = \begin{pmatrix} 0.6 & 0.1 & 0.1\\ 0.1 & 0.8 & 0.2\\ 0.3 & 0.1 & 0.7 \end{pmatrix}$$ I want to find the eigenvalues of this matrix. Because this is a markov matrix, ...
2
votes
3answers
50 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 ?
1
vote
0answers
15 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
0answers
12 views

Two views on a matrix

An n times n matrix $M$ may be viewed as a linear operator on $\mathbb{R}^n$ to itself, but it is also a linear operator on the function space $L^2 (\mathbb{R}^n)$, which is infinite dimensional: ...
0
votes
1answer
37 views

Eigenvalues & eigenvectors of a matrix

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

Use eig and svd syntax in matlab to find complex eigenvalues of a matrix

For matrix $A= \left( \begin{array}{c} 1 & 1 \\ -1 & 1 \\ \end{array} \right) $ when I calculate the eigenvalues (without matlab) , I find $\lambda_1=1+1i$ and $\lambda_1=1-1i$ and when I ...
1
vote
3answers
56 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 a general ...
0
votes
0answers
6 views

Has the degree to which a partial eigensystem of a large sparse matrix approximates the complete eigensystem been determined?

Does anyone know of any studies or results regarding the degree of approximation or the error in estimating the complete spectrum of a large sparse matrix by means of its first $n$ eigenvalues and ...
1
vote
3answers
60 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
24 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 ...
0
votes
1answer
41 views

Unit vector that maximizes or minimizes

I know by the Taylor expansion $f(x,y)$ that in order for the origin to be a minimum point, $f_{xx}$ and $f_{yy}$ have to be both positive. Which I know how to prove. I also know other methods like ...
2
votes
1answer
39 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
23 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 ...
1
vote
1answer
27 views

Is the largest eigenvalue a unique weighted sum of the linear combination of the elements of a matrix?

Let $\lambda$ be the largest eigenvalue of $\boldsymbol{A}\in\mathbb{C}^{n\times n}$ ($\boldsymbol{A}$ is hermitian). Is $$\lambda = ...
0
votes
0answers
10 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
0answers
15 views

Question about eigenvalue problem of a selfadjoint operator.

Let $x=(x_1,x_2)$, and let $X_m$ denote the space of homogeneous polynomial vector fields on $\mathbb{R}^2$ of degree $m$. For example if $m=2$ a vector field $U\in X_2$ is of the form $$ ...
0
votes
4answers
48 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 ...
1
vote
1answer
29 views

Square root of a matrix proof

Let $B$ be a real symmetric $2 \times2$ matrix which satisfies: $$\sqrt{B}v_1=\lambda_1v_1$$ $$\sqrt{B}v_2=\lambda_2v_2,$$ where $v_1,v_2$ are eigenvectors of matrix $B$ and ...
3
votes
1answer
94 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
59 views

Prove that S is diagonal

Let $S: V\rightarrow\ V$ be an operator on an $n$-dimensional real vector space with an eigenvalue that has geometric multiplicity equal to $n-1$. Prove that $S$ is diagonal. Give an example of such ...
0
votes
1answer
39 views

Linear Algebra Problem Proof

I have been stuck on this problem for quite some time now and, unfortunately, appear to have given up. Perhaps the minds on this page will help me out. Given an $n\times n$ matrix D, where ...
0
votes
2answers
26 views

Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
0
votes
0answers
25 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
53 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
28 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 ...
0
votes
1answer
31 views

Showing that $M$ and $N$ will have same eigenvalues.

Today I came accross to this problem. And after some study, I have derived the following solution. Request to the experts, kindly let me know if I have made any mistakes. The question is: if $M, N$ ...
0
votes
0answers
31 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. ...
0
votes
2answers
27 views

Difference of Positive Semidefinite Matrices

Suppose I have two matrices: $$ A\succeq 0\\ B\succ 0 $$ and I know that $$ \langle v_i,Bv_i\rangle - \lambda_i \geq 0 $$ for every normalized eigenpair $(v_i,\lambda_i)$ of $A$. Is this enough to ...
2
votes
2answers
54 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 ...
1
vote
2answers
30 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 ...
1
vote
0answers
39 views

smallest and largest eigenvalue of discretized operator $-d^2/dx^2$

In 1D, the second order derivative operator $-d^2/dx^2$ can be discretized as, using Matlab ...
0
votes
1answer
34 views

Determinant and generalized eigenvalues

Let A, B be two symmetric positive-definite matrices. Let $\lambda_i$ be the generalized eigenvalues of the pencil (A,B). Can we write function $\log\frac{|A|}{|B|}$ (where $|\cdot|$ stands for ...
1
vote
0answers
33 views

Computations for LDA: Eigendecomposition

While reading the book Elements of Statistical Learning p. 113, the author used eigendecomposition of the covariance matrix $\hat{\Sigma}_k =\mathbf{U}_k\mathbf{D}_k\mathbf{U}_k^T$ where ...
0
votes
1answer
38 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 ...
0
votes
0answers
21 views

Ortonormal basis of unitary operator and its spectral decomposition - check my solution.

Dear fellow mathematicians, I'm trying to do a linear algebra exercise, but I have no idea whether I have a correct plan of solution. Here is the problem: Find orthonormal eigenbasis (not sure if ...
4
votes
1answer
88 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 ...
0
votes
1answer
56 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
30 views

Need verification - Given a Hermitian matrix and two eigenvectors corresponding to distinct eigenvalues, show x and y are orthogonal.

Claim: Let $A \in \mathbb{C}^{mxm}$ be hermitian ($A = A^*)$. If $x$ and $y$ are eigenvectors corresponding to distinct eigenvalues, then x and y are orthogonal. Proof: Let $x$ and $y$ correspond to ...
1
vote
2answers
18 views

Continuity in finding eigenvectors

I'm wondering whether there's "continuity" in the eigen vectors of different matrices corresponding to appropriate eigenvalues. For instance, if we change certain elements in a matrix, can we ...
3
votes
1answer
70 views

3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...
0
votes
2answers
45 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
40 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
89 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
25 views

Define a positive dot product in $\mathbb{R^3}$

Consider the matrix $A= \begin{bmatrix} k & k-1 & 0 \\ 1-k & 2-k & 0 \\ 2k-3 & 2k-1 & 2 \end{bmatrix} $ with $k \in \mathbb{R}$ and let be $f_a: \mathbb{R^3} \rightarrow ...