0
votes
0answers
7 views

Calculating the eigenvalues of this matrix

I have the following matrix asociated to a $f:R⁴\rightarrow R⁴$ endomorphism: $\left( \begin{array}{cccc} 1 & b & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & a & 1 \\ 0 & ...
0
votes
2answers
14 views

Linear Algebra, geometric multiplicity

I have a matrix and the question says I that I have an eigenvalue of 0. The question asks me to find the geometric multiplicity of that eigenvalue. I know the answer is 4. I just don't ...
1
vote
2answers
20 views

Diagonalization problems (eigenvalues and vectors)

I am trying to diagonalize the following matrices: $$A = \begin{pmatrix}0 & 1\\-1 & 2\end{pmatrix}\qquad B = \begin{pmatrix}1 & 2\\-1&-1\end{pmatrix}$$ For matrix $A$, I find an ...
0
votes
1answer
18 views

Given 2 matrices generate a reducible algebra, show they have a common eigenvector

Two matrices A, B generate an algebra.... the span of all words made with A and B... example of an element of the algebra: $A^kB^nA^m + I + B^sA^q$ etc... (exponents are all nonnegative). This algebra ...
1
vote
1answer
25 views

Find $P$ such $P^{-1}AP = kR$.

Let $ A=\begin{bmatrix} 3 & -5 \\ 1 & -1 \\ \end{bmatrix}$ Find P such that $P^{-1}AP = k R$ where $k \neq 0$ is a scalar and R an element of $SO(2)$ = {R element of ...
1
vote
0answers
9 views

Resulting Covariance Matrix $\Sigma$' after reducing space along the primary eigenvector?

I am writing a quick & dirty C program to find the first three eigenvectors of a quite large system of points with 512 feature dimensions each. Data is all real. I find the first eigenvector ...
0
votes
0answers
16 views

Transform an almost positive definite matrix to positive definite matrix.

Matrix A (n by n) is constructed as followed. $A(i,i)= -\sum^{k=n}_{k=1}A(i,k), k \neq i. $ A is symmetric. All the off-diagonal elements are negative except two elements $A(p,q)$ and $A(q, p)$: ...
1
vote
2answers
29 views

Calculating the Eigenvectors and Eigenvalues of this Matrix Polynomial

For the matrix $$ A=\begin{pmatrix} 1 & 1 & 2 \\ 0 & -2 & 0 \\ 0 & 2 & 3 \end{pmatrix} $$ How are the eigenvalues and eigenvectors of the following matrices calculated? ...
0
votes
1answer
26 views

Cannot find eigenvectors

How can I find eigenvectors of the following matrix? $$ \begin{matrix} 4 & 0 \\ 0 & 1 \\ \end{matrix} $$ Systematic approach would be: 1. Finding eigenvalues ...
2
votes
2answers
106 views

Is it possible to diagonalize a singular matrix?

I have not seen anywhere written that it is impossible, but it seems impossible, so I want to check if I missed something. According to a theorem, an nxn matrix is diagonalizable if it has n ...
1
vote
4answers
67 views

Proof that $e^x$ is the eigenvector or the derivative operator

I remember hearing my professor talk about how $e^x$ shows up in all our differential equations because it is the eigenvector for the derivative operator. Can someone explain and prove this to me? I ...
1
vote
1answer
27 views

Finding Eigenvalue for cubic equation

I'm learning finding eigenvalues. I learned how to find simplistic eigenvalues for $3\times3$ matrix. By using below way. With this way I can only solve if I have simple determinant equation, like ...
0
votes
1answer
19 views

Projection on cone of non-negative definite matrices

Ok, so if you have a real symmetric matrix $Q$ then the projection of that matrix on the cone of symmetric non-negative definite matrices $\mathcal{C}$ can be explicitly found if we do an ...
6
votes
2answers
62 views

A quick way to estimate eigenvector/eigenvalue of a matrix

Is there a quick way to give a raw estimation of an eigenvector/eigenvalue of a matrix? By "quick" I mean some method which can be computed without a computer or paper and pencil...something you could ...
0
votes
3answers
34 views

For two p.d. matrices $A$ and $B$, prove that $\lambda_1(AB)\leqslant \lambda_1(A) \cdot\lambda_1(B)$

If $A$ and $B$ are two nxn positive definite matrices, then show that $$\lambda_1(AB) \leqslant \lambda_1(A) \cdot \lambda_1(B),$$ where $\lambda_1(\cdot)$ denotes the largest eigenvalue.
-1
votes
1answer
20 views

Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5

Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5 . Then, there exists a non-zero vector $v$ in $R^2$ such that (a) $||Av||$ > 2$||v||$; (b) $||Av||$ < 1/2$||v||$; ...
2
votes
1answer
46 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
16
votes
10answers
1k views

Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
0
votes
1answer
17 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
2
votes
1answer
28 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
1
vote
2answers
28 views

Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is ...
0
votes
1answer
45 views

Is there a significance to a matrix having an eigenvector equal to a column vector within a matrix?

Consider matrix $A$: $\begin{bmatrix} 2 & -2\\ 3 & -3\\ \end{bmatrix}$ After little computing, we find the eigenvectors (and their corresponding eigenvalues) to be equal to $E_{\lambda=0}= ...
3
votes
4answers
188 views

If all eigenvalues are 1 or -1, is then $A^{12}=I$?

True or false: If all the eigenvalues of A are either $\lambda=1$ or $\lambda = -1$ then $A^{12}$= I If we have a matrix $$\mathbf A = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ this has ...
4
votes
4answers
204 views

Every n × n-matrix A with real entries has at least one real eigenvalue. [duplicate]

I have a true/false question: Every n × n-matrix A with real entries has at least one real eigenvalue. I am thinking that this is true but I would like to hear ...
1
vote
2answers
38 views

Finding real, distinct eigenvalues for arbitrary constants

Let $A= \begin{bmatrix} 1 & 1 & 0 \\ -4 & -3 & 1 \\ k & 0 & 0 \end{bmatrix}$. Find all values of $k$ such that $A$ has three real distinct eigenvalues. I have obtained the ...
0
votes
1answer
38 views

Problem determining eigenvalues of a Hermitian matrix

Suppose that you've got an $n \times n$ irreducible matrix $A$ with strictly positive real entries and eigenvalues $\lambda_i$, $i=1,...,m$, arranged so that $|\lambda_1| > \cdots > ...
2
votes
2answers
33 views

Definition: Eigenvalues of a matrix

1) Can a non-square matrix have eigenvalues? Why? 2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible. Thank you!
0
votes
1answer
30 views

Matrix multiplication and rank reduction? - What is the minimal polynomial?

given is a matrix A with $\begin{pmatrix} a & 1 & 0 & \cdots & 0 \\ 0 & a & 1 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 ...
3
votes
3answers
52 views

Besides being symmetric, when will a matrix have ONLY real eigenvalues?

I realize that when a matrix is symmetric, then it must have all real eigenvalues. However, I am doing research on matrices for my own pleasure and I cannot find a mathematical proof or explanation ...
1
vote
1answer
20 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
0
votes
2answers
27 views

Eigen Value & Eigen Vector Pairwise Relationship

Having same eigen values implies eigen vectors are linearly dependent. But why does it not imply that the eigen vectors are same? Are the eigen value and eigen vector pairs not unique for non-zero ...
0
votes
0answers
22 views

Find eigenvalues using matrix representation?

Let $V$ denote the space $P_2(t)\subset \Bbb R[t]$ of polynomials with real coefficients of degree at most $2$. Let $L : V \to V$ be the linear operator given by: $$ L(f(t)) = (t-1)\cdot ...
0
votes
0answers
13 views

It would be possible to use the covariance matrix $C'=XX^T$ instead of the standard $C=X^TX$ to get the same result on PCA?

It would be possible to use the covariance matrix $C'=XX^T$ instead of the standard $C=X^TX$ to get the same result on PCA? If so, what are the next steps to retrieve the data with reduced ...
2
votes
1answer
38 views

Find Eigenvalues of multiplied Matrices when the corresponding Eigenvalues are known

I am trying to find the eigenvalues or in particular the largest eigenvalue of a transformation which consists of two matrices: $A = B C$. Assuming I know the EV of both matrices $B$ and $C$, is ...
0
votes
1answer
15 views

How many iterations are generally required when using the power iteration method?

Suppose I have an n x n matrix and I want to find the dominant eigenvalue and its associated eigenvector. Given these dimensions, what is the minimum number of iterations of the power iteration ...
3
votes
1answer
22 views

Determine the dimension of a vector space.

Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^p(x-b)^q$,where $a$ and $b$ are distinct real numbers. Let $V$ be the real vector space of all $n \times n$ matrices ...
2
votes
1answer
62 views

Conditions for “$AA^T=A^TA$ implies $A$ symmetric” to hold.

This claim arose in this question Show that $A$ is symmetric, with $A \in M_n(\mathbb R)$ where it is assumed additionally that $AA^TA$ is symmetric. I'm considering weakening hypotheses. Let $A$ be ...
2
votes
2answers
101 views

Matrix with no real eigenvalues

Given an $n \times n$ matrix $A$ with real entries such that $A^2=-I$. Prove that A has no real eigenvalues. We can easily prove the following additional statements about $A$ by taking determinants ...
2
votes
0answers
45 views

Standard symmetric tridiagonal matrix Eigenvalue decomposition algorithm?

Hi I am trying to generate an arbitrary Gauss quadrature rule by using the Golub-Welsh algorithm (here). I need to code this on C++ for my personal project. This algorithm involves the eigenvalue ...
1
vote
3answers
37 views

Help on finding eigenvalues of transformation on matrices

T is linear transformation working on 2x2 matrices: T(A) = $\begin{bmatrix}1 & 1\\1 &1\end{bmatrix}$ A as far as I see only 0 is an eigen value but someone told me 2 is eigen value too and ...
0
votes
1answer
41 views

Show that AB and BA have same eigenvalues

If $A$ and $B$ are $n\times n$ matrices, with $A$ nonsingular prove that $AB$ and $BA$ have the same set of eigenvalues. Can we begin with $ABX_1=\lambda_1X_1$, somehow show by manipulation that ...
2
votes
1answer
55 views

Are eigenspaces unique?

I have calculated an eigenspace of a matrix. It is 2 dimensional. I checked it with WolframAlpha, but in WolframAlpha's solution one basis vector in this eigenspace is different from my solution.
2
votes
1answer
58 views

Eigenvalues of $ADA^T$

Consider a rectangular matrix $A\in\mathbb{R}^{M\times N}$ and a diagonal matrix $D\in\mathbb{R}^{N\times N}$. What can one say on the eigenvalues and eigenvectors of $ADA^T$? For example, if we ...
0
votes
1answer
24 views

Eigenvectors of linear transformation

Assume that a linear transformation $T$ has two eigenvectors $x$ and $y$ belonging to distinct eigenvalues $\lambda$ and $\mu$. If $ax + by$ is an eigenvector of $T$, prove that $a=0$ or $b=0$.
0
votes
2answers
32 views

Eigenvalue of a linear transformation

Let $V$ be the linear space of all real polynomial $p(x)$ of degree $\leq n$.If $p \epsilon V$, define $q=T(p)$ to mean that $q(t)=p(t+1)$ for all real $t$. Prove that $T$ has only the eigenvalue $1$. ...
2
votes
0answers
24 views

Lower bound on the smallest eigenvalue

Recently I encountered a lower bound on the minimum eigenvalue of positive Hermitian matrices in (Lower bound on smallest eigenvalue of (symmetric positive-definite) matrix). The lower bound is stated ...
0
votes
1answer
47 views

Why is $ det(A - \lambda I) = (-1)^n \cdot [\lambda^n + c_1\lambda^{n-1} + … + c_n ] $?

Well the title tells you everything I want to know. Why is $ \det(A - \lambda I) = (-1)^n \cdot [\lambda^n + c_1\lambda^{n-1} + ... + c_n ] $ ? With this I then want to show that $ \det(A - \lambda ...
0
votes
1answer
30 views

Question about eigenvalues: eigenvalue $f^2 + f = -1 \rightarrow$ eigenvalue $f^3 = 1$

I have to proof: Let $f \in End(V)$. Show that if $f^2+f$ has eigenvalue $-1$ then $f^3$ has eigenvalue $1$. My idea: If $-1$ is the eigenvalue of $f^2+f$ then there exists (per definition) a $v ...
0
votes
0answers
42 views

Problem with Matrices and Eigen Vectors. Show V1 and V2 are eigenvectors of A.

Suppose A is asymmetric N*N matrix with eigenvectors V, $V=1,2,3...N$ with corresponding eigenvalues X, $X=1,2,3...N$. Pick any two corresponding eigenvectors and eigen values and call them $X_{1}$, ...
0
votes
3answers
38 views

Eigenvalue of a special matrix

A = \begin{vmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{vmatrix} I want to find the eigenvalue of this matrix A. I know how to find its eigenvalues by using rule of ...