0
votes
1answer
34 views

The geometric multiplicity

By given this matrix: \begin{pmatrix}0&a&0\\0&0&1\\0&0&0\end{pmatrix} Why for any a which is not 0 the geometric multiplicity = 1? and why for a = 0 the g.m. = 2? I don't ...
1
vote
3answers
45 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
2
votes
3answers
318 views

Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
1
vote
1answer
46 views

Trace of power of stochastic matrix

I would like to know if this statement is true. Having a stochastic matrix (rows sum up to 1), with a positive (non-negative) diagonal, then it holds that $$\text{trace}({W^2})\leq ...
3
votes
1answer
48 views

$3 \times 3$ real matrix: relation with determinants

$A$ is a $3 \times 3$ matrix with real entries such that $\operatorname{det}(A+I_3)=\operatorname{det}(A+2I_3)$. Then is $2\operatorname{det}(A+I_3)+\operatorname{det}(A-I_3)+ 6 =3 ...
4
votes
0answers
62 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
0
votes
0answers
26 views

Eigenvalue bounds for a positive semidefinite matrix

I have a symmetric $(p\times p)$, positive semi definite matrix $\Omega$. If somebody says: find the eigenvalue bounds of the matrix such that $$w_1I \le \Omega \le w_2I$$ where $I$ is the identity ...
0
votes
1answer
32 views

Equality in the Collatz-Wielandt-formula

Let A be a matrix with positive entries. The perron-frobenius-theorem states that A has a positive dominating simple eigenvalue, called the perron-frobenius-eigenvalue. I denote it with p(A). The ...
3
votes
2answers
68 views

Matrix with all eigenvalues $0$ but not triangular?

Is the situation described in the title achievable? I am looking for a $3\times 3$ case specifically.
-1
votes
2answers
47 views

Matrix with eigen values given find [closed]

Let$$ P=\begin{bmatrix} 0&-2&-3 \\ -1&1 &-1 \\ a&2 &b \end{bmatrix},$$ for some $a,b \in \mathbb{R}.$ Suppose that $1$ and $2$ are eigenvalues of $P$ and $$ ...
0
votes
1answer
56 views

Proof wanted that there is no positive integer matrix with positive integer eigenvalues u,v,w, if $0<u<v$ and $1\le w-v\le 2$

I have the following conjecture : If u,v,w are integers with $0<u<v<w$, then there is a POSITIVE INTEGER 3x3 - matrix A with eigenvalues u,v,w if and only if $w-v\ge 3$. I approved the ...
2
votes
1answer
36 views

Effective way of checking if all eigenvalues of a matrix are integers

Given A matrix with integer entries, it should be checked if all its eigenvalues are integers. Of course, the characteristic polynomial could be calculated, but is there any faster (or easier) ...
2
votes
0answers
32 views

Which n-tuples of positive integers can be the eigenvalues of some matrix with positive integer entries?

This question is closely related to some questions I already asked Given a tuple of positive integers (such as (1,2,5) ), is there a matrix A with positive integer entries such that the integers in ...
3
votes
1answer
54 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
2
votes
1answer
26 views

Lower bound for the spectralradius of a matrix

Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ...
0
votes
1answer
18 views

Difference between matrices with altered eigenvalues

Given two p.s.d. matrices $X_1$ and $X_2$ with eigen decomposition $X_1 = U_1V_1U_1^T$ and $X_2 = U_2V_2U_2^T$ and a constant $\lambda > 0$ Now consider an altered version of the eigenvalue ...
1
vote
1answer
37 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...
1
vote
1answer
29 views

Triangularisation of a linear transformation

I understand that Upper triangular matrices must have at least one eigenvector, but why does this mean that the basis of $[T]_B$ must contain an eigenvector for $[T]_B$ to be upper triangular?
1
vote
1answer
44 views

$A,B$ matrices , prove $Bv = \Lambda v$

$A,B$ are $n \times n$ matrices and $AB = BA$ Also, there is an eigenvalue $\Lambda$ in $A$ which its geometric complexity is $1$. Also there is $ v \ne 0 $ that $v$ is an eigenvector of $A$. ...
-1
votes
0answers
29 views

Order of eigenvectors in jblas?

I am using jblas to compute eigenvectors of a double symmetric matrix. Using symmetricEigenvectors(myMatrix)[0], I can get a matrix which columns are the eigenvectors of my matrix. However I need them ...
1
vote
0answers
85 views

Eigenvalues of $\pmatrix{1&1\\1&2}$

I use maxima for calculation eigenvalues of this matrix: $$ \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix} $$ and I get $\frac{3\pm\sqrt{5}}{2}$ and then $[1,1]$ for some reason. Namely: ...
0
votes
2answers
42 views

Question on Eigen values

Let $A$ be a square matrix and $A^*$ be its adjoint, show that the eigenvalues of matrices $AA^*$ and $A^*A$ are real. Further show that $\operatorname {trace}(AA^*)=\operatorname {trace}(A^*A)$.
1
vote
2answers
64 views

How to find the eigenvalue of matrix A?

We have: $$A\left(\begin{array}{l}\xi \\ \eta\end{array}\right) = \left(\begin{array}{l}a\xi+b\eta \\ a\xi-b\eta \end{array}\right)$$ How to find the eigenvalue of matrix $A$?
3
votes
0answers
95 views

$n$ distinct real eigenvalues of an $n \times n$ matrix

What are the necessary and sufficient conditions for a real $n \times n$ matrix to have $n$ distinct real eigenvalues? Ideally I'm looking for a test that does not require (and is hopefully more ...
2
votes
1answer
33 views

What is the spectral radius of a non-diagonal matrix?

This is my first question in Math StackExchange. Assume that I know the spectral radius of matrix $A$. The matrix $\bar{A}$ is created from $A$ by removing all the $A$'s diagonal entries (i.e., ...
0
votes
1answer
37 views

$T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ and $ <A,B> = Tr(AB^t)$

Let $V = M_{n \times n}(R)$ with the inner product $ <A,B> = Tr(AB^t)$, and $T$ the linear operator given by $T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ . How can i ...
4
votes
3answers
52 views

Questions on Jordan forms

I am studying Jordan form of a matrix from wiki. I am wondering how could two matrices have same eigenvalues with same multiplicities, but have different Jordan form? Also, if two matrices have ...
0
votes
1answer
26 views

Constructing an eigenvector for a certain matrix representing a graph with a perfect code

Let $A$ be a symmetric $(0,1)$-matrix whose row sum is $r.$ Suppose I have a $(0,1)$ vector $v$ such that $$Av = \vec{1} - v.$$ By taking the vector $$u = \vec{1} - (r+1)v$$ we see that $$Au = A ...
1
vote
1answer
35 views

Circulant matrix

$A=\left(\begin{array}{cc} B & C\\ C & B \end{array} \right)$ Here $A$ is the block circulant matrix and B and C are $n \times n$ matrices which are circulant. How can write it as in roots ...
1
vote
2answers
77 views

$\det (A^2 - I) < 0 \Rightarrow \lambda \in (-1,1)$

Let A be real square matrix. If $\det (A^2 - I) < 0$, then A has eigenvalue $\lambda \in (-1,1)$. How to prove this?
1
vote
1answer
32 views

Show that if $A$ is a symmetric matrix with all eigenvalues greater than $0$, then it is positive definite.

Prove that if $A$ is a symmetric matrix with all eigenvalues greater than $0$, then it is positive definite. If $A$ is symmetric then there exists an orthogonal matrix $S$, such that $S^TAS$ is a ...
0
votes
2answers
71 views

Prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$

I want to prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$ If $A\in \mathbb{F}^{m\times n}$ and $B\in \mathbb{F}^{n\times m}$ It is easy to show that $0$ has algebraic multiplicity of at least $m-n$ ...
1
vote
1answer
36 views

diagonalisation unitary matrix

Let $A \in U(n) \subset \mathbb{C}^{n \times n}$ a unitary matrix. Show that: $\exists ~ S\in U(n)$ so that $\bar{S^t}AS=D:=\begin{pmatrix}\lambda_1&&0\\&\ddots & ...
4
votes
1answer
63 views

Eigenvalues of a $4\times 4$ parameters matrix

Let $a,b,c,d\in\Bbb{C}$ and $B =\begin{bmatrix} a & b & c & d\\ d & a & b & c\\ c & d & a & b\\ b & c & d & a\\ \end{bmatrix}$ I ...
0
votes
2answers
35 views

Eigenvectors of a hermitian matrix to the same eigenvalue

Probably, this question has already been answered, but I did not find an answer. If a matrix A is hermitian and an eigenvalue $\lambda$ has multiplicity k, are there always k pairwise orthogonal ...
1
vote
0answers
9 views

Maximum Eigenvalue and a corresponding Eigenvector of an infinite Hilbert matrix

I have the following matrix $$H=\begin{bmatrix} 1 & \frac{1}{2} & \cdots & \mbox{ad}\ +\infty\\ \frac{1}{2} & \frac{1}{3} & \cdots & \mbox{ad}\ +\infty\\ \vdots & \vdots ...
1
vote
1answer
34 views

If A and B are real orthogonal matrices how to prove that either A-B or A+B is singular?

Degree of matrices is odd $n$-th degree. I figured out all eigenvalues of matrices A and B have to $1$ or $-1$. Now I assume I have to prove $\det((A-B)(A+B)) = 0$ and from that either $\det(A-B)$ or ...
1
vote
1answer
32 views

What's the connection between rank of matrix and $0$ eigenvalue?

My matrix B is nxn and know nothing about if diagonalizble, but I know that rank B = 1. Therefore the geometric multiplicity of λ=0 as an eigenvalue is n-1. But by knowing the rank is 1, can I say ...
0
votes
1answer
47 views

Do T and T* have the same eigenvalues with the same algebraic multiplicity?

I know that the eigenvalues of T* are the conjugates of T's eigenvalues , but how can I see each eigenvalue of T and it's conjugate , the eigenvalue of T*, have the same algebraic multiplicity?
6
votes
2answers
467 views

Why integration operator has no eigen values?

Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that ...
0
votes
2answers
43 views
0
votes
1answer
38 views

Is it true that every eigenvalue has at least one eigenvector?

As mentioned above: Is it true that every eigenvalue has at least one eigenvector? Or is it possible that while trying to find the basis of a specific eigenspace, i will get only the zero vector ...
0
votes
2answers
22 views

calculate principal components

suppose that we have following data ...
0
votes
1answer
39 views

real matrices $2\times 2$ and $3\times 3$ that are not similars to a diagonal matrix

Example of real matrices $2\times 2$ and $3\times 3$ that are not similars to a diagonal matrix. I find that $A =\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix} $ then i suppose that its ...
1
vote
3answers
41 views

Prove that if $A$ is invertible then $AA^\top$ is positive definite [duplicate]

I need to prove that if $A$ is a square invertible matrix then $AA^\top$ ($A$ multiply $A$ transpose) is positive definite. I tried to prove that all the eigenvalues are positive. I know that ...
1
vote
5answers
79 views

$A$ a $n\times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$

Let $A$ a $n \times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$. a) Give an example that satisfies this conditions. b) what are the eigenvalues ​​of $A$? Well for $a)$ i ...
3
votes
1answer
22 views

Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$ Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix) Prove: $A$ is ...
1
vote
2answers
47 views

Eigenvalue by inspection?

Can I guess the eigenvalues of a $3\times3$ matrix having all entries $1$? for e.g., consider the matrix $ \left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 ...
2
votes
0answers
19 views

Linearly independent skew symmetric complex matrices having the least eigenvalues

Question: Let $A$, $B$ be two $5 \times 5$ (or $7 \times 7$) skew-symmetric complex matrices (i.e. $A^t = -A$), and suppose that $$ \forall t,s \in \mathbb{C}, \quad M(t,s):=(tA+sB)^*(tA+sB) \text{ ...
0
votes
1answer
40 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...