For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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-2
votes
1answer
36 views

True / False about a matrix

Let $A= \begin {pmatrix} x & y \\ -y & x \end {pmatrix}$ where $x,y \in \mathbb{R}$ such that $x^2+y^2=1$. 1) For any $n \ge 1$, $$A^n= \begin {pmatrix} \cos\theta & \sin \theta \\ -\sin ...
1
vote
2answers
28 views

Eigenvalues of $6 \times 6$ matrix?

Which of {$\pm1,\pm i$} are the eigenvalues of matrix A, $$A=\begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & ...
1
vote
1answer
20 views

$P$ is an matrix invertible Proof $|\lambda I-PBP^{-1}|=|\lambda I -B|$

I have this problem : $P$ is an matrix invertible Proof : $|\lambda I-PBP^{-1}|=|\lambda I -B|$ I'm not so sure about my answer, since I don't think I could use "double" determinant for example ...
3
votes
0answers
70 views

Second structural equations in lorentzian space $\Bbb L^3$.

I'm rewriting O'Neill's "Elementary Differential Geometry"'s section on connection forms in Lorentz-Minkowski space $\Bbb L^3$, and I'm having trouble proving the second structural equations $${\rm ...
2
votes
1answer
39 views

A matrix version of L'Hopital's Rule?

Is there a version of L'Hopital's Rule for matrix calculus? For example: let $A$ be a symmetric $n\times n$ positive definite matrix and $b$ be an $n\times 1$ vector. As $b$ converges to $0_{n\times ...
6
votes
1answer
132 views

Powers of adjacency matrix doesn't seem to correspond to observed number of paths on graph

I would really appreciate some help on this! $A^n$ represents $n^{th}$ power of the adjacency matrix of a graph. I keep reading that the $A^n_{ij}$ entry equals "the number of paths of length n ...
0
votes
1answer
36 views

Proof If $A_{2x2}$ and $\lambda$ real number then $|\lambda I-A|=\lambda^2-(\operatorname{tr}A) \lambda+|A|$

I have this problem : Proof If $A_{2x2}$ and $\lambda$ real number then $|\lambda I-A|=\lambda^2-(\operatorname{tr}A) \lambda+|A|$ This is what I did : I took an arbitrary $A$ $$ A= \left( ...
1
vote
2answers
27 views

Question (true/ false) about the rank of an induced matrix. [on hold]

Let $A$ be a $5\times 5$ matrix and let $B$ be obtained by changing one element of $A$. Let $r$ and $s$ be the ranks of $A$ and $B$ respectively. Which of the following statements are true. $s \leq ...
3
votes
1answer
47 views

Perturbation theory of the eigenvalues about the symmetric matirx

From Weyl's theorem, i.e.: Let $A$ and $E$ be $n\times n$ real symmetric matrices. Let $\alpha_1\geq\ldots\geq\alpha_n$ be the eigenvalues of $A$ and $\hat{\alpha}_1\geq\ldots\geq\hat{\alpha}_n$ be ...
0
votes
1answer
23 views

Rank of a general matrix

Given some scalars $a_1,a_2,...,a_m \in F$ not all zero and $b_1,b_2,...,b_n \in F$ not all zero, what is the rank of the matrix $M=(a_i b_j)_{\begin{matrix}1 \leq i \leq m \\ 1\leq j \leq n ...
2
votes
0answers
27 views

LU factorization of a modify matrix

Suppose you know $L$, $U$, decomposition LU of a matrix $M+I$ ($M+I=LU$). Lets $J$ a diagonal matrix whose elements are $0$ or $1$. Is there any relation between the factorization LU of $M+I$, and ...
0
votes
0answers
44 views

Eigenvalues of $\frac{1}{2}(A+ A^T)$ [on hold]

If we know the eigenvalues of $\frac{1}{2} (A+A^T)$ with $A$ a real $m \times m$ matrix, what can we say about the eigenvalues of $A$?
0
votes
0answers
10 views

Ring structure of tuples mod k

Consider a vector of n integers $$A= a_1, a_2, ... a_n$$ Such that for another vector $$B= b_1,b_2... b_n$$ $$AB^T \equiv 0 \mod k$$ For an integer k. I was playing around with these structures ...
0
votes
0answers
16 views

SVD and base changes matrices

I'm not hugely comfortable with linear algebra, so wanted to double check that the following reasoning was correct. Does it hold that, given two matrices R and B $U R B U^T=U R U^T U B U^T= U R U^T ...
1
vote
0answers
25 views

Prove the Basis of Column Vectors

Let $V$ be a vector space over field $\mathbb{F}$. Let $B=\{u_1,...,u_n\}$ be an ordered basis of $V$. Show that $\{[u_1]_c,...[u_n]_c\}$ is a basis of $M_{n,1}(\mathbb{F})$ for every ordered basis ...
0
votes
2answers
45 views

Prove an upper bound for the determinant of a matrix A

Let $A$ be a $3 \times 3$ real matrix with all $0\le a_{ij} \le 1$. Show that $\det(A) \leq 2$ and find such matrices with $\det(A) = 2$. Let $A$ be a $n \times n$ matrix with all $0\le a_{ij} \le ...
0
votes
1answer
35 views

Dot “power” of a matrix

By analogy with the matrix product is there a name for the matrix "power" operation defined by $$y_i = \prod_j x_j^{a_{ij}}?$$ For example: $$\left( \begin{array}{lll} x_1 & x_2 & ...
0
votes
2answers
62 views

What is the determinant of matrix?

Find determinant of the $n \times n$ permutation matrix $$ M= \left[ {\begin{array}{cccc} 0 & 0 & \ldots & 0 & 1\\ 0 & 0 & \ldots & 1 & 0\\ \vdots & ...
1
vote
1answer
29 views

Schur product theorem

The theorem states that the Hadamard product of two positive definite matrices $ A \circ B$ is also positive definite. Can I make any statement about a the Hadamard product of a positive definite ...
2
votes
1answer
20 views

Prove that $A$ is similar to $B$ probably using Jordan form

Let $A, B \in M_n(\mathbb{F})$ such that: $a_{ij} = 0 \iff b_{ij} = 0$. $a_{ij} = b_{ij}$ for all $i \ne j+1$ $\exists \lambda \in \mathbb{F}$ such that $a_{ii} = b_{ii} = \lambda$. Prove that ...
1
vote
2answers
32 views

Finding inverse linear transformation

I'm solving a homework question and I'm stuck with it's last part. The question goes like this: Let $\displaystyle T:M_n(\mathbb{R})\to M_n(\mathbb{R})$ be a transformation defined as ...
0
votes
0answers
22 views

Cubic 2x2x2 matrices (tensors), is there a special name for them?

Is there a special name for cubix 2x2x2 tensors? What properties an algebra on these matrices has?
1
vote
3answers
60 views

Prove that if $\mathbf A$ is an $n\times m$ matrix, then $\text{tr}(\mathbf A \mathbf A')=\text{tr}(\mathbf A' \mathbf A) $

If $\mathbf A$ is an $n\times m$ matrix, then $\text{tr}(\mathbf A \mathbf A')=\text{tr}(\mathbf A' \mathbf A) \text{ where } \mathbf A'\text{ is transpose of }\mathbf A\text{ and tr}(\mathbf A ...
2
votes
1answer
49 views

Prove that $A$ is similar to $B$

Let $A, B \in M_n(\mathbb{F})$ such that $m_A(x) = m_B(x)$ and $f_A(x)=f_B(x)=(x-\lambda_1)^{d_1}\cdots (x-\lambda_k)^{d_k}$ for different $\lambda_1, \ldots, \lambda_k$ such that $1 \le d_l \le 3$ ...
1
vote
1answer
25 views

Endomorphism ring as a set of matrices

Let $A=\mathbb Z[\sqrt{-5}]$, and let $I=(2,1+\sqrt{-5})$ (which is known to be a non-principal ideal of $A$ with $I^2=2A$). If we put $P=A \oplus I$, my question is: "why the endomorphism ring of ...
0
votes
1answer
17 views

Can $LL^T$ decomposition of a matrix be computed by the same algorithm as $LU$-one?

I know that's the silly question. But if I perform $LU$ decomposition on a symmetric positive definite matrix, will this decomposition be the same one as $LL^T$ one?
1
vote
0answers
32 views

Prove that $V_i$ are $T$-invariant for $1\le i\le k$ and $V=\bigoplus_{i=1}^{k}V_i$

Let $T$ be a linear operator over $V$ with dim$(V)=n$ and let the ordered set $B=${$v_1,v_2,...v_n$} be a basis for $V$. Furthermore, let $A=[T]_B$ be block-diagonal matrix (that is ...
2
votes
7answers
92 views

Prove that a matrix equals to its transpose

Let $A$ be a $(n\times n)$ matrix that satisfies: $AA^t=A^tA$ Let $B$ be a matrix such that: $B=2AA^t(A^t-A)$ Prove/disprove that: $B^t=B$ I started with: $$\begin{align} B &=2AA^t(A^t-A) \\ ...
1
vote
2answers
25 views

What row-operations allow this $\operatorname{Mat}_{2\times2} (\mathbb{R})$

$$ A = \begin{pmatrix} 1 & r \\ s & 1 \\ \end{pmatrix} \Rightarrow \begin{pmatrix} 1 & r \\ 0 & 1-s \cdot r \\ \end{pmatrix} = B \quad\quad r,s \in \mathbb{R} $$ Matrix B is ...
0
votes
0answers
16 views

Matrix operation repeat matrix members

I am going to use C++ Armadillo library which handles matrices to generate matrix $B$ and $C$ from matrix $A$. $$ A=[M_0,M_1,\ldots,M_{n-1}]^T $$ $$ ...
-5
votes
1answer
57 views

Inequality on matrix norm: $ \lVert A^n \rVert \leq \lVert A \rVert^n $ [on hold]

If $A$ is a $n \times n$ matrix and assume we have a matrix norm $\lVert \cdot \rVert$. In a proof I need the following property: $ \lVert A^n \rVert \leq \lVert A \rVert^n $. I don't know how to ...
3
votes
2answers
37 views

At least one diagonal element of any real symmetric matrix of rank $1$ is non-zero ?

If $A$ is a real symmetric matrix of rank $1$ then is it true that at least one diagonal element is non-zero ?
1
vote
2answers
46 views

Jordan form of a matrix

Let $$A = \left( {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & 2 & 0 \cr 0 & 0 & 0 & 3 \cr 0 & 0 & 0 & 0 \cr } } \right)$$ The ...
0
votes
1answer
23 views

Linear transformation to higher dimensional space.

There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds: $ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) ...
-1
votes
0answers
21 views

Differential equation to space state excercise

This is a "back of chapter" excercise which im trying to solve, my answer doesnt match the solution printed on the book, I want to write the equation in state space matrix form without using the ...
2
votes
2answers
28 views

Can a low-rank matrix set have nonempty interior?

The answer to this question may be super simple, but it is very not obvious to me. Consider the space $S^n$ of symmetric $n\times n$ matrices. Consider $T\subset S$ the set of rank $n-1$ matrices. ...
-3
votes
0answers
36 views

Let A,B be nxn matrices such that detA Not equal to 0, but detB = 0: Show [on hold]

Let $A$, $B$ be $n\times n$ matrices such that $\det A \neq 0$, but $\det B = 0$. Show $\|A-B\|_2 \geq(\|A^{-1}\|_2)^{-1}$.
1
vote
1answer
73 views

Let $A$, $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = O$

Let $A$ and $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = 0$. Progress I know that $ABA=0 \implies A^2B=0$. Here ...
3
votes
2answers
35 views

Exponential restricts to special linear matrices

Let's consider a field $k$ of characteristic $p$ and a matrix $M \in \mathfrak{sl}_n$ (the Lie algebra of trace $0$ matrices). Assume $M^r = 0$ for some $r < p$ so that the exponential $$\exp(M) = ...
1
vote
1answer
56 views

Set of linear equations with coefficients - solution using matrices

I have a set of linear equations: \begin{matrix} ax_{1}& {}+bx_{2}& {}+x_{3}& & =0\\ cx_{1}& {}+dx_{2}& &{}-x_{4} & =0\\ & {}-ex_{2}& ...
-6
votes
0answers
31 views

i need help with a matrices [on hold]

Can someone show me how or make me a Matrices with the answers being 7,15,4?
0
votes
0answers
40 views

A matrix transformation from R^4 to R^3 - linear algebra - how to find the image of a point

I'm trying to revise for an upcoming exam on linear algebra and have come across this question. I do not understand the line "the image of a point (x1, x2, x3, x4) can be computed from the defining ...
1
vote
0answers
9 views

name, notation for “block inner product” $X^H Y$

Given a set of $k$ vectors of length $n$, $X = [x_1, \dots, x_k]$ and another set of $l$ vectors of length $n$, $Y = [y_1, \dots, y_l]$, I'd like to to compute the inner product of every combination ...
0
votes
0answers
20 views

Change of base - Hermitic matrices

This exercise comes from a university exam (http://www.ubacs.com.ar/foro/viewtopic.php?f=67&t=3079, link in spanish). I'll copy it in english for everyone. It's #3: We define in $C^{n×n}$ the ...
2
votes
3answers
168 views

Linear dependency of nilpotent matrices

I would like to prove that four $2\times 2$ nilpotent matrices are always linearly dependent, using the Cayley-Hamilton theorem or the minimal polynomial in some way. I think I have proved the ...
0
votes
7answers
118 views

Given matrix P such that $P^{102 } =0 $ , to show that $P^{2} = 0$.

P is given to be a 2×2 matrix such that $P^{102} = 0$. How to show that $P^{2} =0 $?
1
vote
0answers
16 views

Probability measure of rank-$r$ matrices

I have a question about the distribution of matrices with a specific rank. Consider $\mathcal{M}^{n\times m}$ the set of all $n \times m$ matrices with entries in some field $\mathbb{K}$. If I define ...
1
vote
2answers
54 views

Using inverse of matrix A as approximate inverse of matrix that is very close to A

Say we have two matrices, $A$ and $A'$ so that $A\approx A'$, and we have the inverse of $A$, $B$, where $AB=I$, and the inverse of $A'$ where $A'B'=I$. If we have some guarantee about how big any ...
0
votes
1answer
17 views

Null/Col/Row space be a line\plane through the origin?

For a $4\times3$ matrix can the nullspace, the column space and row space all be lines through the origin? For a $2\times4$ matrix can the nullspace, the column space and row space all be planes ...
-1
votes
2answers
43 views

Nullspace, row space, column space in $m\times n$ matrices [on hold]

For a $4\times 3$ matrix can the nullspace, the column space and row space all be a line through the origin? For a $2\times 4$ matrix can the nullspace, the column space and row space all be a plane ...