0
votes
3answers
18 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.
0
votes
1answer
24 views

On the eigenvalues / properties of a specific matrix.

I'm not sure how to better phrase the title of the question, because I don't know the specific name of the matrix I am after, but I want to consider matrices of the form $$ \begin{align*} ...
2
votes
2answers
53 views

Showing that matrix admits an eigenvector?

Let A= a b c d be a 2 x 2 matrix, where a,b,c and d are real numbers. We say that A admits an eigenvector if there exists a unit vector u and a real ...
1
vote
2answers
51 views

When do eigenvectors converge?

Let $A_n$ be a sequence of self-adjoint $N\times N$ matrices that converge in the operator norm to $A$. The sequence of eigenvalues of $A_n$, denoted $\lambda_n$, converges to an eigenvalue of $A$, ...
2
votes
1answer
29 views

How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

I have the following matrix $$ \begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix} $$ First I got the eigenvalues which are $0$, $1$, $2$. I tried to get the ...
0
votes
0answers
6 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
2
votes
1answer
45 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 ...
0
votes
1answer
25 views

Any transformation in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$. [on hold]

$V$ is a two-dimensional vector space over a field $\mathbb F$; prove that every element in $A(V)$, the ring of linear transformations on $V$, satisfies a polynomial of degree $2$ over $\mathbb F$.
2
votes
1answer
25 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 ...
4
votes
1answer
33 views

What are eigenvalues of higher order finite differences matrices?

I know (at first empirically, then read somewhere) that for for second order finite differences matrices like $$\begin{pmatrix} -2&1&0&0&0\\ 1&-2&1&0&0\\ ...
1
vote
1answer
44 views

Find an $n\times n$ integer matrix with determinant $1$ and $n$ distinct positive eigenvalues

I feel pretty stupid for doing this, but here goes anyway. Earlier today I asked: Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues. As it turns out, for my problem I ...
1
vote
0answers
22 views

3-Species Population Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
4
votes
2answers
65 views

Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
0
votes
1answer
44 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}= ...
0
votes
0answers
17 views

Low rank matrix square root

I'm trying to perform canonical correlation analysis (CCA) between matrices $X$ ($n \times p$) and $Y$ ($n \times k$), with covariance matrices $S_{X}=XX^T/(n-1)$ and $S_{Y}=YY^T/(n-1)$ respectively, ...
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?
1
vote
1answer
26 views

Why are eigenvectors of an invertible matrix linearly independent?

Question asked in the title! Thanks guys
0
votes
1answer
31 views

Finding Eigenvalues with Gershgorin-Discs

$A=\begin{pmatrix}-5 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & -5 & 4\end{pmatrix}$ Find the Gerschgorin-discs, where the eigenvalues of $A$ lie. According to the formula if we ...
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 ...
1
vote
1answer
18 views

Prove that adjacency matrix has negative eigenvalue

We are given non-oriented graph without loops. Task is to prove that adjacency matrix of that graph has negative eigenvalue. I put some effort into drawing a proof here , but it seems that I'm ...
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 ...
1
vote
1answer
25 views

Does this question make any sense - eigenvalues and norms

Im having difficulties understanding this question: show that if $b$ is an eigenvector of an invertible matrix $A$ with an eigenvalue $\lambda_1$ and $\delta b$ is an eigenvector of $A$ with an ...
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
0answers
21 views

How to ensure a matrix of a special rank

As described in the subject, how can I ensure a matrix of a special rank. for example, given matrix A of m*n and m>n; Then, how can I mathematically constrain the matrix A to be rank n? As we all ...
-2
votes
1answer
26 views

Finding eigenvalues and eigenvector [closed]

A is a matrix, 0 1 2 2 0 0 0 0 1 Find the eigenvalues of A and for each eigenvalue find the corresponding eigenvector. I have no clue for the whole ...
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 & ...
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
37 views

the solution of matrix polynomials

In order to get the eigenvalues of \begin{equation} P=\left[ \begin{array}{cc} 0_{n\times n} & I_{n\times n} \\ -A & -B% \end{array} \right], \end{equation} where $A$ and $B$ are both $n\times ...
4
votes
2answers
62 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 ...
0
votes
1answer
26 views

Proof of Lyapunov Stability for Constant Matrix System

I am trying to find the necessary and sufficient conditions for the point of equilibrium x=0 of $x'=Ax$ to be Lyapunov stable, where A is constant matrix. The book I'm using briefly touches on this, ...
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 ...
0
votes
0answers
30 views

find out complex eigenvalues by looking at matrix

is it possible for a matrix to have complex eigenvalues by just looking at the matrix? if the matrix has all positive distinct numbers, ie |1 4 7 13| |2 5 8 12| |3 6 9 11| |1 5 9 10| also is ...
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
69 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 $A$ ...
1
vote
0answers
49 views

Maximizing the product of first Eigenvalues of rank-1 hermitian matrices

Suppose we have $L$ complex vectors $\mathbf{a}_{l}$ with dimension $N\times 1$ I want to solve this optimization problem $\mathbf{x}_{\mathrm{opt}}=\arg ...
1
vote
1answer
55 views

Eigenvalues less than or equal to 1

What proprieties does a square $n\times n$ real matrix $\mathbf M$ need to have in order to have all it's eigenvalues be less than or equal to one in absolute value? I'm looking for proprieties such ...
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 ...
0
votes
2answers
30 views

Proof that Jordan form with eigenvalues of 0 is nilpotent

I can see that if I have a nxn matrix Of Jordan form with eigenvalues of zero $\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}$ How do I prove that $A^n = 0$ ...
12
votes
0answers
169 views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
1
vote
2answers
23 views

Calculating Null Space: Xn = 0

So, I am calculating the null space of bases and matrices in order to get eigenvectors, and I occasionally come across matrices which will have a row like the following: (Put in system format) $x_2 = ...
0
votes
1answer
21 views

Lower bound on the minimum eigenvalue of sum of two matrices

Assume that $A$ is a symmetric positive definite matrix and $B$ is a symmetric (can potentially negative entries). Is the following bound correct? $$\lambda_{min}(A+B)\geq ...
2
votes
1answer
68 views

If $A$ and $B$ are symmetric, $A^5=B^5$ then $A=B$

Let $A$, $B$ be real symmetric $n\times n$ matrices such that $A^5=B^5$. Prove that $A=B$. Here are my attempts: The following identity holds $(A-B)(A^4+A^3B+A^2B^2+AB^3+B^4)=0$ and yields at least ...
0
votes
1answer
43 views

Properties of a matrix and eigenvalues

A, B, C are three real-square matrices. A is an upper triangular matrix with all of its diagonal entries equal to zero. B is a matrix such that $b_{ij}=-b_{ji}$, and C is a matrix such that $\sum_j ...
1
vote
1answer
28 views

Largest eigenvalue of a special m-matrix

How to estimate the largest eigenvalue of followed characteristics? Let $A={a_{ij}}$. Symmetric positive definite. Real. Very sparse. Diagonal elements are all positive, and off-diagonal elements ...
5
votes
3answers
63 views

A question on eigenvalues

Let $A,B\in M_{2}(\mathbb{R})$ so that $A^2 = B^2 = I$. Which are eigenvalues of $AB$? 1) $1\pm \sqrt 3$ 2)$3 \pm 2\sqrt2$ 3)$\dfrac {1}{2},2$ 4)$2 \pm 2\sqrt 3$
4
votes
3answers
43 views

Show non-symmetric matrix has non-orthogonal eigenvectors

I'm struggling with a problem from Boas's Mathematical Methods in the Physical Sciences. The question is, for a 2x2 matrix M s.t. M is real, not symmetric, with eigenvalues real and not equal, show ...
0
votes
1answer
36 views

Rank-one modification of graph Laplacian

Suppose I have a Laplacain matrix for a 3-node-path graph as follows $L=\left[\begin{array}{ccc} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{array}\right]$ Now, I want to ...