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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0
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2answers
51 views

Finding all matrices for which the homogeneous system has a given solution space

Find all $3\times 3$ matrices for which the homogeneous system has a solution space as the line $x = 2t$, $y = t$, $z = 0$. (Hint: Write the row reduced augmented matrix from given information.) ...
0
votes
3answers
45 views

Rank of matrices and their product

Let $\operatorname{rank}(A_{3 \times 3})=\operatorname{rank}(B_{3 \times 3})=2$. I need to figure out whether $AB=0$ is possible. On the one hand, I know that $\operatorname{rank}(AB) \leq ...
3
votes
2answers
72 views

Given $A$ is $6×6 $ real symetric matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $

Given $A$ is $6×6 $matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $. I knowthat rank of matrix doesnot change when we square it , but how to proceed in this question.Any hints ? Thanks
5
votes
4answers
319 views

Proof If $AB-I$ Invertible then $BA-I$ invertible.

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
0
votes
0answers
21 views

How to compute the unique positive eigenvector 'v' in Analytic Hierarchy Process

I'm trying to calculate the values in the right most column v but I have absolutely no idea how to do it. I've done some prelim work and managed to get pretty much everything else in the table in ...
1
vote
3answers
36 views

Matrix multiplication ambiguity

From this source here, it says that matrix multiplication is given by this: $AB = \begin{bmatrix} a_{1,1}b_{1,1}+a_{1,2}b_{2,1}+...+a_{1,n}b_{p, 1} & ...\\ \vdots & ...
0
votes
0answers
22 views

Multiplications of non-square matrices and the dependencies of row vectors.

I'd like to find $D$ and $L$ for a given $H$. $H$ is a 7-by-6 matrix. Its rank is 6. All sub-matrices of $H$ are full rank. In other words, if we choose any $n$-by-$n$ sub-matrix within $H$, where $n ...
0
votes
1answer
75 views

How do row operations affect the column space?

I've been curious about this: Row operations do not affect the row space, but they affect the column space. Is there any way to 'systematically' perform row operations to make the column space the ...
4
votes
1answer
71 views

Matrix equation solution

Does anybody know how to solve this matrix equation: $$P = P P^T R + X,$$ where $P, R,$ and $X$ are vectors with $n$ elements, and $P$ is the unknown vector?
0
votes
1answer
24 views

Relationship between type of matrix and eigenvalues

Prove that if the eigenvalues of a diagonalizable matrix $A\in M_n(\mathbb{R})$ are all $1$ or $-1$, then $A^{-1}=A$ What I tried to reverse the way to get the rough idea. $$A^{-1}=A\implies ...
-1
votes
1answer
64 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
44 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
38 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
141 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 ...
1
vote
0answers
89 views

A question on matrices

Let $M\in\Bbb \{0,2\}^{n\times n}$ be a rank $t\leq n$ matrix and we know that it can be rewritten as $A+B$ where $A$ has $\{0,1\}$ entries and symmetric and $B$ has $\{-1,0,+1\}$ entries and skew ...
2
votes
1answer
63 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 ...
8
votes
1answer
436 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
50 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( ...
4
votes
1answer
118 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
31 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
53 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
29 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
23 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
33 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 ...
1
vote
2answers
88 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
42 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
76 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
59 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
23 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
47 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
28 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
72 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
62 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 $1 \le d_l \le 3$ ...
2
votes
1answer
50 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
21 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
44 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
114 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
32 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
20 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 $A$. $$ A=[M_0,M_1,\ldots,M_{n-1}]^T $$ $$ ...
3
votes
2answers
55 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 ?
2
votes
2answers
65 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
37 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) ...
2
votes
2answers
32 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. ...
1
vote
1answer
81 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 ...
4
votes
2answers
46 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) = ...
2
votes
0answers
62 views

Inequality proof critique

I'm trying to prove an inequality. I think I'm correct but it would be helpful if my proof can be critiqued. Given two matrices of same dimension $T$ and $P$ with $P \geq 0$ and scalar $\delta>0$, ...
1
vote
1answer
60 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}& ...
1
vote
0answers
15 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
29 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
200 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 ...