0
votes
1answer
20 views

Does eigenvalues bounded below imply matrix norms bounded below?

If we have a sequence of matrices $A_n$ such that all of the eigenvalues are positive and bounded away from $0$, is it true that $|A_nx| \geq \lambda|x|$ for some $\lambda>0$? Thank you
1
vote
1answer
15 views

Eigen Values and Nature of Matrix [on hold]

Let J be a 3x3 matrix all of whose entries are 1.Then (i)0 and 3 are the only eigen value of J (ii)J is positive semi definite. (iii)J is diagonalizable (iv)J is positive definite.
1
vote
2answers
16 views

Matrix norm question.

When do we know that $|Ax| \geq \lambda|x|$ for all $x$ where $A$ is some matrix and $\lambda$ is some constant > 0. Is it enough if all of the eigenvalues are positive? If so, can you please prove ...
1
vote
3answers
21 views

Transponse and inverse

If I have two lower diagonal matrices L and M If I have $(L^TL)^{-1} (M^TM) = I$ where $L^T$ is the transpose of L can I say $(LM^{-1})^T (LM^{-1}) = I$ why? and using what lemma or theorem.. ...
0
votes
0answers
26 views

Connecting inner products and the trace of a matrix

Hello, I'm currently trying to work through this question. However, I am struggling with understanding what I should do. I am aware of all of the definitions used throughout, however I cannot link ...
2
votes
1answer
17 views

number of eigenvalues = dimension of eigenspace

Is this true in general? What about: number of negative eigenvalues = dimension of span(eigenectors for the negative eigenvalues)? Or even more generally: number of eigenvalues greater than 4.3 = ...
1
vote
1answer
27 views

Show that $[A,\exp(B)]=\exp(B)[A,B]$

Denote $\exp(A)=\sum_{k=0}^{+\infty} \frac{A^n}{n!}$ where $A\in M_n(\mathbb{R})$ and $[A,B]=AB-BA$ Assume that $A,B$ commute with $[A,B]=AB-BA$ Show that $$[A,\exp(B)]=\exp(B)[A,B]$$ ...
-1
votes
1answer
47 views

chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
0
votes
2answers
31 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
2
votes
3answers
38 views

Finding matrix with respect to given bases

Given that A: \begin{matrix} a & b & c \\ d & e & f \\ \end{matrix} is a matrix of T : V -> W with bases G = {g1, g2, g3} and Q = {q1, q2}, respectively. Find the matrix of T ...
2
votes
1answer
41 views

If a set of 2x2 matrices are independent, do they also span M22?

I am looking for a basis in M22. So I need to get a linearly independent set that also spans the vector space. I have worked out how to tell independence, but I am stuck on the spanning requirement. ...
0
votes
1answer
16 views

Singular matrices over a commutative ring $R$, with a given adjoint matrix

First, I apologize if this is a duplicate question. I also must apologize if this has a trivial solution. This question has two parts: Let $R$ be a commutative ring with $1$, and let $F = R^n$ be a ...
1
vote
2answers
21 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 ...
2
votes
3answers
70 views

Showing Orthogonality

How would I do this question..... I'm familiar with Gram-Schmidt and the basics but I have no idea how to do $a$ and $b$ in this question. Suppose $\{\vec x_1, \vec x_2, \vec x_3\}$ is an ...
0
votes
1answer
19 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 ...
4
votes
5answers
54 views

Find the axis of rotation of a rotation matrix by $INSPECTION$ (NOT by solving $Kv=v$)

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$ by INSPECTION. This is from my other thread ...
1
vote
1answer
15 views

Similar Matrices Conditions:

Sorry this question was already asked but my english is not good. For matrix to be similar, does it have to have all of these properties or SOME of them? Same determinant Same Trace Same ...
0
votes
1answer
17 views

Prove that the $j$-th column of $AB$ is the product $Ab_j$

Prove that the $j$-th column of $AB$ is the product of $A$ and the $j$-th column of $B$ First of all, THIS IS NOT HOMEWORK. This was a homework. I can prove this using the fact that $e_j$ extracts ...
1
vote
1answer
24 views

Is every invertible matrix over an algebraically closed field diagonalisable?

In $\Bbb{R}$ the only invertible matrices (I can think of) that are not diagonalisable are those which stand for a rotation, but in $\Bbb{C}$ this shouldn't be a problem anymore, since rotations can ...
0
votes
0answers
19 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 $A(q, p)$: ...
3
votes
0answers
31 views

Does the cross section of $[-1,1]^n$ on a $k$-dimensional subspace always contains a rotated image of $[-1,1]^k$?

This question is inspired by a recent bounty question, but the two questions are different and solving this one, I believe, will not lead to an answer of that bounty question. Suppose $n>k\ge1$ ...
3
votes
1answer
30 views

What is a reducible algebra?

In my matrix analysis book, a set of complex matrices is said to be an "algebra" if 1)it is a subspace, 2)whenever A and B are members, so is AB. Then it uses the terms reducible and irreducible ...
0
votes
2answers
24 views

How to find the matrix of a transformation relative to standard basis?

Given $b_1=(-1,3)$ and $b_2=(1,-2)$ which make a basis for $\mathbb R^2.$ If $$ T(b_1) = 6b_1 + 7b_2 \quad\text{and}\quad T(b_2) = 3b_1 + 8b_2, $$ find the matrix of $T$ relative to the standard basis ...
1
vote
2answers
30 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? ...
2
votes
3answers
48 views

Explain why $S$ is not a basis for $\mathbb{R}^3$

Explain why $S$ is not a basis for $\mathbb{R}^3$ $S=\{(1, 3, 0),(4, 1, 2),(-2, 5, -2)\}$ I set this equal to an arbitrary vector $\mathbf{x} = (x_1, x_2, x_3)$ After solving I got the matrix: ...
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
107 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 ...
0
votes
1answer
21 views

Find the Axis of rotation of rotation matrix $K$ after solving $(K-I)v=0$

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$. This is from my previous thread click here ...
1
vote
0answers
18 views

Normality of the product of a diagonal matrix and an SPD matrix?

I believe this to be true, but can't seem to prove it exactly: suppose $A$ is symmetric positive definite, and $D$ is a diagonal matrix. Then, $A$ is diagonal if $DA$ is normal for any diagonal ...
3
votes
1answer
27 views

Determinant of a matrix with symmetric positive definite block

In reviewing linear algebra for an exam, I encountered the following problem: Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive definite. If $x$ is any nonzero vector, show that $$ ...
4
votes
1answer
91 views

determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
0
votes
1answer
31 views

matrix logarithm, determination and trace maxmimzation

Let $A = A^\ast \in M_n$ be a positive definite matrix ($\lambda_i(A) > 0$). Show that $\log\det(A)-Tr(A)$ is maximized by $A = I$.
1
vote
1answer
78 views

Relative Eigenvalue Perturbation Bound deduction from Ostrowski's Theorem

I need to deduce the relative eigenvalue perturbation bound from Ostrowski's Theorem. In short i need to proove ´this statement; $\frac{|\lambda_k(SAS^*)-\lambda_k(A)|}{|\lambda_k(A)|} \leq ...
1
vote
1answer
55 views

Hermitian Matrix with their eigenvalues arranged in non-decreasing order

I need to formulate one property of Hermitian Matrices. It goes like this; If A, B $\in M_n$ are hermitian and their eigenvalues are arranged in non-decreasing order , then $\lambda_i(A+B)\leq ...
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 ...
0
votes
2answers
37 views

Find the axis of rotation from the rotation matrix.

This is a problem from the book "Mathematical Methods in the Physical Sciences" Third Edition by author Mary L. Boas. on page 129, Example 5, just in case any of you are familiar with it. So I ...
1
vote
0answers
32 views

Mapping unit sphere to ellipsoid

Consider an $N$-dimensional space. Let $M$ be a square $N\times N$ (real, but I am interested in complex case too) matrix. Are the following (hyper)ellipsoids (or degenerate hyperellipsoids)? $\{v ...
0
votes
1answer
65 views
+100

Matrixes and modulo of a vector

Consider an $N$-dimensional space. Consider the function $\kappa$ which maps a square $N\times N$ matrix $M$ into the scalar field $v\mapsto \lvert Mv \rvert$ (for $v$ being a vector). Is the ...
1
vote
0answers
23 views

Gerschgorin Theorem singularity proof

I know how to prove the Gerschgorin Theorem but how exactly would one show that there are no values of $\mu$ s.t. $\mu<0$ for which $A-\mu B$ is singular where $$ A= \begin{bmatrix} ...
0
votes
1answer
39 views

Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues - Proof Strategy [Lay P397 Thm 3]

Herein, I denote the Hermitian conjugate by * (ie: $A* = \bar{A}^T) $. Let $v_i$ and $v_j$ be two eigenvectors of an Hermitian matrix H. First of all suppose that their respective eigenvalues i and j ...
1
vote
1answer
51 views

Minimzing the generalized dissimilarity measure

I am trying to solve the following problem for quite some time now, but with no progress. Here is the problem. Let $x_1....x_n$ be n samples in d-dimmensional space and let $S$ be a non ...
1
vote
1answer
39 views

Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
0
votes
2answers
103 views

Proof: $Ax=x$ for all $x$ implies $A=I$ [on hold]

Let $A$ be a square matrix of order $n$ and let $x$ be an $n$-vector. Prove that if $Ax=x$ for all $x$, then $A=I$. Thanks in advance
1
vote
4answers
83 views

Prove the matrix satisfies the equation $A^2 -4A-5I=0$ [on hold]

How to prove that $$ A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} $$ satisfies the equation $A^2 -4A-5I=0$?
2
votes
2answers
52 views

solving $X^2 - 3X - A = 0$ where $A,X \in \mathbb{M_2(\mathbb{R})}$

Given $A = \begin{pmatrix} 7 & 3 \\ 3 & 7 \end{pmatrix}$ find a $2\times 2$ matrix $X$ s.t. $X^2 - 3X - A = 0$, in the previous parts I have diagonalised $A$ and got $P^{-1}AP = \begin ...
2
votes
0answers
28 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
2
votes
1answer
67 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
2
votes
0answers
38 views

linear binary code problem

Let $\mathcal C$ be a $[n,k,d]$ linear binary code such that $\mathcal C$ has a systematic generator matrix $G=[I_k\mid A]$. (i) Prove that $u\in (\mathbb F_2)^k$ is coded by $c=(u\mid uA)\in ...
1
vote
2answers
39 views

Standard matrix A of T?

Help please. What would be the standard matrix of A? I know how to do number 2 and 3 but I'm just having trouble with A. I asked this earlier but I lost my account and I'm not sure if I posted ...