2
votes
1answer
41 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 ...
13
votes
9answers
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
12 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
23 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 ...
0
votes
2answers
19 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
42 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
186 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
194 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
33 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
36 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
32 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
48 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
16 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
26 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
12 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
34 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
21 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
59 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
99 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
36 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
36 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
39 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
54 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
22 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
29 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 ...
0
votes
2answers
38 views

Eigenvalues of negative companion matrix

Here's a homework question I've been stuck on for a while. Given $A = \left[ \begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & a_0 \\ -1 & 0 & 0 & \cdots & 0 & ...
3
votes
1answer
44 views

Find a formula in terms of k for the entries of Ak, where A is the diagonalizable matrix:

Find a formula in terms of k for the entries of $A^k$, where A is the diagonalizable matrix This is my 2x2 matrix (sorry for formatting): [$-5$ $8$] [$-4$ $7$] I've tried this question a million ...
1
vote
1answer
33 views

Linear Algebra: Eigenvectors and eigenvalues

Find a basis for the corresponding eigenspace to the listed eigenvalue: $$A=\left[\begin{matrix}4&0&1\\-2&1&0\\-2&0&1\end{matrix}\right], \lambda=1$$ This is what I've come up ...
2
votes
1answer
123 views

Simple question about inequality involving eigenvalue of a matrix and a scaled version of that matrix

Let $x$ be a vector in $\mathbb{R}^d$. Let $A^*=\sum_1^{i=n} x_ix_i^T$ and $A=\sum_1^{i=n} a_ix_ix_i^T$ for $a_1>a_2>\cdots>a_n$ for $i<j$ where the second summand is called $A^*$. Is ...
4
votes
2answers
61 views

Determinant of rank-one perturbation of a diagonal matrix

Let $A$ be a rank-one perturbation of a diagonal matrix, i. e. $A = D + s^T s$, where $D = \DeclareMathOperator{diag}{diag} \diag\{\lambda_1,\ldots,\lambda_n\}$, $s = [s_1,\ldots,s_n] \neq 0$. Is ...
3
votes
0answers
48 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
2
votes
2answers
50 views

Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can ...
3
votes
4answers
107 views

How to find characteristic polynomial of this matrix?

Let, $A=\begin{bmatrix} 4&0&1&0\\1&1&1&0\\0&1&1&0 \\0&0&0&4 \end{bmatrix}$. Knowing that $4$ is one of its eigenvalues, find the characteristic ...
1
vote
2answers
39 views

Eigenmatrices of a given vector.

Given a vector $v$, I would like to find the set of its "eigenmatrices" - that is, the set of all matrices $A$ s.t. $Av=\lambda v$ for some constant $\lambda$. (Following this, I would like to ...
1
vote
1answer
23 views

Does the order of Q matter?

I have found the eigenvalue and from that I have found the eigenspaces of A. The next step is to find orthonormal eigenvectors. The problem has three different eigenspaces. When I was solving the ...
0
votes
2answers
30 views

Given a matrix $A$ with eigenvalue…

Given a matrix $A$ with the eigenvalue $ \lambda $ and eigenvector $v$. Let $b$ be some vector. Show that the vector $v$ is also an eigenvector for the matrix $B = A-v b^T$ and construct a formula for ...
4
votes
2answers
68 views

Show each eigenvalue of a companion matrix has geometric multiplicity $=1$.

Given the differential equation $$x^{(n)}(t)+c_{n-1}x^{(n-1)}(t) + \dotsb + c_1x'(t) + c_0=0,$$ we can form a vector $\xi = (x, x', \dotsc, x^{(n-1)})$, and then we have $$\xi'(t) = A\xi,$$ where $A$ ...
2
votes
1answer
80 views

$AB=BA$. Prove $B$ is diagonalizable.

Question: A and B are matrices size $n\times n$ given $AB=BA$ and A has n eigenvalues, prove $B$ is diagonalizable. I would have written what I tried to do, but It's really nothing worth reading.. ...
1
vote
1answer
48 views

Eigenvalues of symmetric matix

Well for a symmetric matrix ($A^T = A$), is there an easy algorithm to get the eigenvalues by hand? Especially for moderate sized matrices like 6*6 (where calculating/solving the determinant becomes ...
0
votes
1answer
52 views

Find eigenvalues from a given relation.

This is a simple problem of linear algebra. One without knowing graph theory may solve it. I am missing a small easy logic. Description: Let $G$ be a graph with $n$ vertices and $G^c$ is its ...
1
vote
1answer
39 views

Operator Norm = 1

Let there be a linear map $T$ such that $T: \mathbb R^n\to\mathbb R^m$. The operator norm $\lVert \cdot\lVert_{op}$ of $T$ is then defined as the largest value of $c$ for which $\lVert T(\vec v)\lVert ...