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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1answer
22 views

How to write this as one matrix?

I have a rather easy question, but I do not come along. Let $Y$ be a $(n\times 1)$-matrix, $A$ a $(n\times n)$-matrix and $B$ a $(k\times n)$-matrix. Consider $X=(AY,BY)$. How can I write $X$ in ...
0
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0answers
14 views

Variance with Matrices

I have a problem in calculating the variance that is described in this paper : http://www.actuaries.org/LIBRARY/ASTIN/vol32no1/171.pdf The variance should be a single number. I have never calculated ...
0
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1answer
25 views

Matrices in systems of linear equations

I've been working on matrices lately. Currently, I am stuck on solving systems of linear equations using matrices. I've read the following article which has proved very helpful in understanding the ...
0
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1answer
29 views

Find a $S$ base in $\Bbb R^3$ such as $[T]_S=A$

let linear transform $T$, $T:\Bbb R^3\to \Bbb R^3$ $(x,y,z)\to(-3x+y+4z,2x-3z,y+5z)$ find a base $S$ in $\Bbb R^3$ such as $ [ T]_S=\begin{pmatrix} 1&-2 &2 \\ 3 & 1&5 \\ 2 & ...
4
votes
1answer
331 views

Gaussian Matrix Integral

I need your help to solve this exercise : Let $S$ be a symmetric Hermitian matrix $N\times N$ : $S=(s_{ij})$ with $s_{ij}=s_{ji}$. When $\langle s_{ij}s_{kl}\rangle\neq 0$ What $$\int ...
2
votes
1answer
116 views

Proof that frobenius norm is a norm [duplicate]

It's pretty basic and I'm sure I'm missing something dumb here, but I'd like to know why $||A+B||_F \leq ||A||_F+||B||_F$ The way I understand it, ...
2
votes
2answers
34 views

A result on extension fields in linear algebra.

Let $F$ be a subfield of $E$, $A$ an element of $\mathcal{M}_F(m,n)$ and $b$ a vector in $F^m \subset E^m$. What is the easiest way to prove the following statement: if $Ax = b$ has a solution in ...
2
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0answers
27 views

Ordering binary matrices for reflection/rotation

I have a collection of $n\times n$ binary matrices and I would like to reduce it for symmetry ($D_4$ -- reflections and rotations). The naive method of testing each pair is very slow because the ...
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1answer
86 views

For which $x,y,z,w$ is matrix $A$ orthogonal/unitary?

given is $A = \frac{1}{2} \begin{pmatrix} x & 1 & 1 & 1 \\ y & 1 & -1 & 1 \\ z & 1 & -1 & -1 \\w & 1 & 1 & -1 \end{pmatrix} $ How do I have to chose ...
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1answer
39 views

2-Norm of a complex matrix equation

I am having trouble understanding the following excerpt from a math text I'm working through: My question specifically is how line 2 came about in the expansion. How do the real and imaginary parts ...
2
votes
0answers
51 views

matrix inverse and limit

I would like to get a better understanding of limits and matrix inverses, specifically the relationship between: $\lim_{k\rightarrow \infty}(\mathbf{A}^{-1})$ and $(\lim_{k\rightarrow ...
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1answer
156 views

Find bases given that P is the change of coordinates matrix from this to this [Lay P244 Q4.7.19]

Lay P289: Let $V$ be an $n$-dimensional vector space, let $W$ be an $m$-dimensional vector space, and let $T$ be any linear transformation from $V$ to $W$. To associate a matrix with $T$, choose ...
4
votes
1answer
291 views

Characterization of positive definite matrix with principal minors

A matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, for ...
0
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1answer
19 views

how to represent mathematically a matrix with distinct values

I would like to write a mathematical expression for the following matrix A = (5 7 3 9 1 8 2 6 4) in words, the matrix "A" contains the values from 1 to ...
1
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1answer
69 views

Computing eigenvalues of principal submatrices of Kronecker product of two PSD matrices

Given two PSD matrices $A \in R^{n \times n}$ and $B \in R^{m \times m}$ with eigenvalues $\lambda_i$ and $\mu_j$ respectively, the eigenvalues of the Kronecker product $A \otimes B$ are given by ...
4
votes
1answer
123 views

$AB-BA=A$ implies $A$ is singular and $A$ is nilpotent. [duplicate]

Let $A$ and $B$ be two real $n\times n$ matrices such that $AB-BA=A$ Prove that $A$ is not invertible and that $A$ is nilpotent. My attempt is the following. It holds that $AB=(B+I)A$ If ...
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3answers
46 views

“$1$ is an eigenvalue of $A^n$” implies an eigenvalue of $A$ is a root of unity?

Let $A$ be a square matrix. If $1$ is an eigenvalue of $A^n$, then is it true that there is an eigenvalue of $A$ which is a root of unity?
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2answers
44 views

Function for diagonalizing a vector.

I was playing around whith the idea of what operation (function) should I perform (apply) over a vector $\mathbf{a} = (a_1,a_2, \ldots, a_N)^T \in \mathbb{R}^N$ to come up with the following matrix: ...
1
vote
1answer
40 views

represent hankel matrix by low rank tensorial approximation

suppose that we have given following matrix \begin{matrix} x_1 & x_2 & ..x_p \\ x_2 & x_3 & ...x_{p+1} \\ . & .& . & \\ x_{N-p+1} & x_{n-p+2} &... x_n ...
2
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2answers
208 views

Rotate Existing Vector

Hello and apologies if the title of the question is not very precise. Question: I am reading the document talking about the simulation of photons in tissues using a Monte Carlo simulation. The exact ...
0
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1answer
41 views

What is the derivative of a skew symmetric matrix?

I'm trying to work out some Jacobians and I ran across a problem. If I have a function of a vector making it a skew symmetric matrix, like below, what is the derivative $f'$? $$ ...
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2answers
55 views

When is a companion matrix diagonalizable and what does this say about the associated field extension?

Consider the $n\times n$ matrix $$ M=\begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & -c_0\\ 1 & 0 & 0 & \cdots & 0 & 0 & -c_1\\ 0 & 1 & 0 & ...
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2answers
43 views

Method to Multiply many Matrices simultaneously

Imagine you have three random matrices $$ A = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right] ; \quad B = \left[ \begin{array}{cc} 3 & 4 \\ 5 & 6 \end{array} \right]; ...
2
votes
1answer
130 views

Is the matrix least squares minimizer (Frobenius norm) the same as the matrix 2-norm minimizer?

Given matrices $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{n \times k}$, consider the (least squares) minimizer $\arg \min_{X \in \mathbb{R}^{m \times k}} \| AX - B \|_F$, where $\| M \|_F$ ...
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2answers
81 views

Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
2
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0answers
20 views

Find the coordinates of the matrix after a reflection in the given line.

$$\left[\begin{matrix}-8 & 1 & -7\\ -7 & -5 & 1\end{matrix}\right]$$ The given line is the y axis. I cannot show my work for I have no clue on how to solve this.
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1answer
48 views

Calculating eigenvalues

I need to calculate the characteristic polynomial and eigenvalues of the following matrix. It's been a long time since my linear algebra courses, so I have pretty much lost the ability to compute such ...
2
votes
0answers
64 views

kernel space of linear combination of matrices

Suppose $A$ and $B$ are $N\times N$ matrices so that for every $x$ and $y$, $xA+yB$ has a kernel of dimension at least $2$. Is it necessarily true that $\ker(A)\cap\ker(B)$ has dimension at least ...
0
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0answers
88 views

Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P95]

This question is not a duplicate of the original, in which user Shuchang proved the question. Presently I'm asking about further intuition or a picture, and no proofs please. $1.$ Intuitively, in ...
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3answers
80 views

How to tell if a linear system is consistent

So I have a list of equations and have made it into REF which gives me $$\left[\begin{matrix}1 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0\end{matrix}\middle|\begin{matrix}1 \\ ...
2
votes
1answer
87 views

Eigenvalues and Eigenvectors for matrix. Complex Eigenvalues

How can I find out the eigenvectors for this matrix: $$A= \begin{pmatrix} -3 &0&0\\ 0&3&-2\\ 0&1&1 \end{pmatrix} $$ I found the eigenvalues: $\lambda_{1}=-3$, ...
0
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0answers
18 views

What is the difference between closed and open subsets in reducibility of a graph

I've read somewhere the following sentence, the graph A is reducible to at two closed subsets. Is that different than just saying "A is reducible" ?? What is the difference between open and closed set ...
3
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0answers
95 views

How can we solve this question without brute force

If $A\in GL_n(R)$, where $R$ is a commutative ring with identity, I would like to prove $$ M=\begin{pmatrix} A & 0 \\ 0 & A^{-1} \\ \end{pmatrix}\in ...
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0answers
22 views

The second eigenvalue of a reducible stochastic matrix

The magnitude of the second dominant eigenvalue of a reducible matrix, as I know, is supposed to be 1, why it's not the case for this matrix : $$ \begin{matrix} 0 & 1 & 0 ...
0
votes
2answers
29 views

What can you say about the range space and null space

Let $V$ be a vector space over a field $F$ and $T$ a linear operator on $V$. If $T^2$$=$ $0$, what can you say about the relation of the range of $T$ to the null space of $T$?
2
votes
1answer
28 views

Spectral norm of a matrix obtained by setting some entries to zero

For example can we say, that if $A$ is original matrix and $A'$ obtained from $A$ by zeroing some elements then $\|A\|_2 \geq \|A'\|_2$?
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2answers
26 views

Change of Base matrices

Let $E$ and $F$ be two bases of the same n dimensional vector space $U$ $\bullet$ if $P$ is the change of base matrix from $E$ to $F$ and Q the change of base matrix from $F$ to $E$ then ...
0
votes
1answer
39 views

How to calculate a Frobenius norm?

Suppose that $A$ is an $n \times m$ ($n$ less than $m$) full rank matrix. Apply Gram-Schmidt orthogonalization to the rows of $A$, then we get an $n \times m$ matrix $B$ with orthonormal columns. ...
2
votes
2answers
70 views

How do I prove this matrices question?

Let $$ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & \omega^2 & \omega\\ 1 & \omega & \omega^2 \end{pmatrix}$$ where $\omega \ne 1$ is a cube root of unity. If ...
2
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0answers
28 views

Multiplication of matrices [duplicate]

When we add two matrices we just simply add the corresponding elements but when we multiply two matrices there is a much more complex process.Why does it happens?
1
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1answer
39 views

Matrix multiplication: $X_{r \times c}$ and $Y_{c \times d}$

Matrix $X$ has $r$ rows and $c$ columns, and matrix $Y$ has $c$ rows and $d$ columns, where $r, c$, and $d$ are different. Which of the following must be false? The product $YX$ exists The product ...
5
votes
2answers
207 views

Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in ...
1
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1answer
132 views

Linear Algebra matrix notation

My question is referring to the following $4 \times 6$ matrix: $$\begin{bmatrix} 0 & 1 & 2 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 ...
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2answers
67 views

Eigenvalues of $A+B$ in this special case

Let $A$ and $B$ are real, square matrices with the same dimension. We know that $\text{rank } A = 1$ and we know the eigenvalues of $A$. Furthermore, we know that $B$ has only zeros in the diagonal, ...
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0answers
44 views

Two person zero sum problem, help/guidance needed..

I'm a computer science student and I have this problem I need to solve for my games theory course. I don't have an example to follow, or use as guidance, and my colleagues are not very helpful( as in, ...
2
votes
2answers
89 views

How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
1
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1answer
63 views

Hessian matrix as derivative of gradient

From a text: For a real-valued differentiable function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, the Hessian matrix $D^2f(x)$ is the derivative matrix of the vector-valued gradient function $\nabla ...
0
votes
1answer
16 views

Understanding the basis term

Consider: $$\left( {\matrix{ 0 & 1 & 2 \cr 0 & 0 & 0 \cr } } \right)$$ I want to find a basis for the row-space of the matrix above. One might say $$B = \left\{ {\left( ...
0
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2answers
277 views

How do I show this

Given invertible matrices $A,B$ and $P$ such that $A = PB$, then we say that $A$ is left equivalent to $B$. Show that left equivalence is indeed an equivalence relation.
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1answer
43 views

Prove: $R(A+B) \subset R(B)+R(A)$

Prove: $R(A+B) \subset R(B)+R(B)$ If it's not clear $R(A)$ is the the row-space of $A$. Let $(A+B)_i$ the $i$-th row of $(A+B)$. We can write it as a linear combination of $A$ and $B$. ...