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
71 views

A Matrix Optimization Problem

Given an $n\times d$ matrix $Y$, I am looking for an algorithm to find an $n$-vector $\mathbf{v}$ ($0\le \mathbf{v}_i\le 1$ for all $i$) that minimizes $\sum_{i:X_i<0}X_i$, where $X:= \mathbf{v} ...
1
vote
2answers
46 views

If we add $I$ to a matrix $M$, does that mean we always add 1 to each of $M$'s eigenvalues?

Title says it all, Suppose we have a matrix $\mathbf{M} \in \mathbb{R}^{N \ \text{x} \ N}$, with eigenvalues $\lambda_i$, for $\ i = 1, 2 ... N$. If we now add the identity matrix $\mathbf{I}$ to ...
0
votes
1answer
38 views

Product rule type formula for $\nabla \cdot (M(x)v(x))$ where $M(x)$ is a matrix and $v(x)$ is a vector?

Let $M(x)$ be a $n\times n$ matrix with each element depending on $x$ a variable on $\mathbb{R}^n$. Let $v(x)$ be a vector. Is there a nice product rule formula for $\nabla \cdot (M(x)v(x))$?
1
vote
1answer
27 views

Linear transformation over real matrix spaces

I got the following problem: Let $S:\mathbb{M^R}_{3 \times 3} \to \mathbb{M^R}_{3 \times 3}$ be a linear transformation defined by $S(A) = (3A+A^T)/2$ for every matrix $A \in \mathbb{M^R}_{3 \times ...
3
votes
1answer
84 views

Is this determinant identity true?

I simulated the following $$\det(I+[A|B][A|B]^*)\geq\det(I+[B][B]^*)$$ and every time I get a true result. So how can I prove this statement? Here $[A|B]$ is matrix augmentation. $I$ is the identity ...
6
votes
3answers
426 views

Number of matrices with no repeated columns or rows

If you consider all $10$ by $15$ matrices with entries that are either $0$ or $1$, there are ${2^{15} \choose 10}$ with no repeated rows (up to row permutation) and ${2^{10} \choose 15}$ with no ...
4
votes
1answer
69 views

Show that if $tr(A+B) > tr(A)$ then $tr((A+B)^k)\geq tr(A^k)$ for any $k\geq 1$

This may be a stupid question, but I am completely stuck, I don't even know where to start. I have to show that if $tr(A+B) > tr(A)$ then $tr((A+B)^k)\geq tr(A^k)$ for any $k\geq 1$, where $A$ and ...
1
vote
1answer
86 views

Linear Algebra Span question

Let $a, b, c$ be vectors in $\mathbb{R}^3$. From what I understand, if $c\in \mathrm{Span}\{a,b\}$, then $b\in \mathrm{Span}\{a,c\}$. Since they all fall on the same plane, I can't seem to find a ...
2
votes
1answer
25 views

Factorizing a block column matrix with element-wise factors

Is it possible to factor this matrix $$\begin{bmatrix} x_{11} a_{11} & x_{11} a_{12} & x_{12} a_{11} & x_{12} a_{12} & \\ x_{21} a_{21} & x_{21} a_{22} & x_{22} a_{21} ...
0
votes
2answers
37 views

Linear Algebra Matrix Question solutions

Hi I was just wondering if an augmented matrix had no pivot positions, would the system have infinite solutions? Since it has no pivot positions that means, the columns must be filled with 0s and it ...
1
vote
2answers
146 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...
0
votes
1answer
34 views

The entries of a random matrix

I am confused about something. Whenever I create an $n\times n$ random matrix using Matlab (using the command $A=\mathrm{rand}(n,n)$), I get a square matrix whose entries are all between $0$ and $1$. ...
0
votes
2answers
65 views

Vector space and Dual space

I'm struggling with this problem: Let $V$ be a vector space over a field $F$ and let there be $l_1,l_2 \in V^*$. I need to show that if $l_1(x)l_2(x)=0$ for every $x \in V$ then at least one of ...
0
votes
1answer
40 views

Two different ways to write C(A)?

let $\mathrm A \in \Bbb R^{m\times n}$ I know that the three fundamental subspaces are: $\mathrm \ker(\mathrm A) = \{ x \in \Bbb R^n : \mathrm Ax = 0 \} = \{x\in \Bbb R^n : \langle ...
1
vote
2answers
343 views

Combine a rotation matrix with transformation matrix in 3D (column-major style)

I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. I want this rotation matrix to perform a rotation about the X axis (or YZ plane) ...
6
votes
1answer
92 views

Eigenvalues appear when the dimension of the Prime Index Matrix is a prime-th prime. Why?

I had a look at the eigenvalues of the matrix, I called it Prime Index Matrix (is there a better name?), constructed like the following: $$ P_{k,p_k}=P_{p_k,k}=1, $$ where $p_k$ is the $k$th prime. ...
1
vote
1answer
228 views

Derivative of matrix inverse w.r.t. vector

I need to differentiate the inverse of the $K\times K$ symmetric matrix $A$ w.r.t some vector (that $A$ depends on). Is there a rule for this? In case I do the derivative w.r.t. to some scalar there's ...
0
votes
1answer
75 views

Inequality matrix norm

Let $A$ be an $n\times n$ random matrix $A=rand(n,n)$. Let $\alpha=max_{i,j}|a_{ij}|$ (i.e, $\alpha$ is the largest entry in $A$ in absolute value).I need to show that $\ \alpha < \| A \|_{2}$. ...
1
vote
1answer
2k views

Need help in understanding how to find an elementary matrix

I read this chapter in my book and thought I understood it, but I don't. I tried working a problem to test my understanding and I just don't know how to get started. Given the following matrices: ...
1
vote
1answer
74 views

Getting stonewalled on computation of $2\times 2$ Hessian matrix

The question: Let $z \in R^N$, and let $f(z) = \log[1^T z] \in R$. I am told that the Hessian matrix of this function is the following: $$ H = \frac{1}{1^Tz}\Big[ 1^Tz \mathrm{diag}(z) - zz^T \Big] ...
0
votes
1answer
30 views

Understanding matrix equations

If I have two matrices $ A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 0 & -3 & -2 \\ \end{bmatrix} $, $ AX = \begin{bmatrix} 2 & ...
0
votes
1answer
104 views

Determinant of a symmetric, positive semidefinite, sparse integer matrix

I'm looking for an algorithm that calculates the (log) determinant of a symmetric, positive semidefinite, sparse integer matrix. Does such an algorithm exist that can exploit both sparsity and ...
-1
votes
1answer
81 views

Product of matrices: is $A\times A=0 \implies A = 0$ true or false? [closed]

Is the statement $A \times A = 0 \implies A = 0$ (where $A$ is a square matrix) true or false? If it is true, prove it. If it is false, give a counter-example. Edited I did not know about nilpotent ...
2
votes
0answers
28 views

properties of row reduction to explain $Ax - b = 0$ being true

How would i go about using properties of row reduction to explain why the equation $Ax - b = 0$ is true? I am not sure how to attack this. I know that $Ax=b$ where $b$ is a linear combination of the ...
0
votes
2answers
796 views

determine whether the equation $Ax = b$ is consistent for every $b$ in $\mathbb R^m$

I have two problems, the first one is the following matrix: $$\begin{bmatrix}1 & 0\\ -2 & 1\end{bmatrix}$$ where the RREF is $$\begin{bmatrix}1&0\\0&1\end{bmatrix}$$ and where the ...
3
votes
1answer
228 views

Linearly independent functionals

Let $ f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...
0
votes
1answer
59 views

Matrix norm of product equal implies equality in norms of factors

Given a matrix $A$, if $$\|Av\|_1=\|Aw\|_1$$ for given vectors $v$ and $w$, then does $\|v\|_1=\|w\|_1$? Here $\|\,\cdot\,\|_1$ denotes the $L^1$ norm.
2
votes
1answer
100 views

intuition for matrix multiplication not being commutative

I want to have an intuition for why A*B in matrix multiplication is not same as B*A. It's clear from definition that they are not and there are arguments here (Fast(est) and intuitive ways to look at ...
2
votes
1answer
90 views

Commuting matrices still commute if you conjugate one of them?

If two square matrices, $A$ and $B$, of the same size and with complex entries commute, why does $\overline{A}$ (the complex conjugate of $A$) commute with $B$? If this is not true, is it true if one ...
1
vote
1answer
158 views

Relation between condition and linear dependence of column vectors

There are several interpretations of the condition number of a matrix: Relation between smallest and largest singular values, amount of error amplification etc. In my opinion, another interpretation ...
1
vote
1answer
42 views

Finding an odd determinant

I'm looking for the determinant of a rather odd looking matrix, for scalars $x_i,y_i$ with $ 1\leq i \leq n$ the matrix $A_{n\times n}$ defined this way : $A_{ij} = x_i\cdot y_j$ except for when $i=j$ ...
-5
votes
1answer
70 views

Describe the Transpose of $A= \left[\matrix{0 & 0 & 1\\0 & 1 & 1\\ 1 &1 &1} \right]$

$$A= \left[\matrix{0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 0& 0& ...
0
votes
1answer
119 views

Find all real values of $\lambda$ such that $b$ is a linear combination of $a_1, a_2, a_3$

Let $a_1 = (3,2,5), a_2 = (2, 4, 7), a_3 = (5,6,\lambda)$. Find all real values of $\lambda$ such that $b = (1,3,5)$ is a linear combination of $a_1, a_2, a_3$. I'd be glad if you could ...
0
votes
3answers
109 views

Why is this 'obviously' positive semi-definite?

Here is a snapshot from a book I am studying. I learned all about positive semi-definiteness, and in fact I know that this matrix they are showing is in fact PSD. What I do not know is how they ...
1
vote
1answer
88 views

Linear Algebra Help! Matrices with respect to given basis

Hey Guys I'm new here so I dont know much exactly how to do all the fancy symbols but here it goes: Let $V_n$ be the vector space of real polynomials of degree at most $n$ and Let $B_n$ be the usual ...
2
votes
1answer
32 views

Set of Matrices and Invertibility

If $S= \{A_1,\cdots ,A_k\}$ is a set of row equivalent matrices, and there's a linear combination of the elements of $S$ which makes an invertible matrix, then I need to show that every one of the ...
1
vote
2answers
62 views

$A^{-1}BA=$? Easy Matrix Algebra

I have constant matrices $A$,$S$ & $B$ $$(A'SA)^{-1}A'SB$$ Can I simplify to $(A'A)^{-1}S^{-1}SA'B$, where $S^{-1}S = I$ and therefore drops out?
0
votes
1answer
48 views

What is the order of this group? [duplicate]

Let $H$ be the subgroup of the group $G$ of all $2 \times 2$ non-singular matrices whose entries are integers modulo a given prime $p$ consisting of those and only those matrices in $G$ whose ...
2
votes
2answers
154 views

Show that $A$ and $A^{-1}$ have same eigenvalues?

If $A$ is a square matrix of order $2$, and determinant of $A$ is $1$, then prove that $A$ and its inverse have the same eigenvalues. So, let $\lambda_1$ and $\lambda_2$ be the eigenvalues of $A$. ...
1
vote
2answers
800 views

Consider the trace map $M_n (\mathbb{R}) \to \mathbb{R}$. What is its kernel?

The map is the trace map. I.e, it takes any $n$ by $n$ matrix and associates to that matrix, a number of the form $\mathrm{Tr}(A) = \sum_{i=1}^n a_{ii}$, where $A \in M_n (\mathbb{R})$. I need to ...
1
vote
3answers
372 views

Find a three independent vectors u, v, w that each lie in N(A), the null space of A.

Let $A=\begin{bmatrix}0& 0& 0& 0 \\ 3& 9& 3& 9\end{bmatrix}$. How should I figure this out? I know the first column has the Pivot and the other three columns have free ...
2
votes
2answers
102 views

What is the structure of matrix multiplication and minus?

Please note I have only little background im mathematics and I am working on formalizing theorems with theorem provers. This is very much a beginner question. Suppose I have matrices, where the ...
0
votes
2answers
48 views

Relating determinant of two matrices

Consider a symmetric square matrix $g$ of dimension $N$ and another symmetric square matrix $h$ of dimension $n$. Suppose $S$ is a $N\times n$ matrix such that $$ h = S^T g S $$ Suppose $\det g \neq ...
1
vote
0answers
87 views

Will the projection of a singular matrix into an orthonormal space be non-singular?

I'm working through an implementation of the solution from 16.3.1 Dealing with the nullspace in the case of a singular within-class scatter matrix when performing discriminant analysis. In this ...
2
votes
2answers
78 views

How to prove this matrix bound

Let an $m$ by $n$ matrix $A\in\mathbb C^{m\times n}$. Denote $M=\max_i\sum_{j=1}^n|A_{ij}|$ and $N=\max_j\sum_{i=1}^m|A_{ij}|$. Prove for any two vectors $x\in\mathbb C^m$ and $y\in\mathbb C^n$, we ...
0
votes
1answer
210 views

Differentiation of bilinear form w.r.t. matrix

I need to do a derivative of bilinear form: b'C a w.r.t to Kx1 vector t where "b" and "a" are Kx1 vectors and "C" is KxK matrix that depends on vector t (and a and b are independent of t). Does anyone ...
0
votes
4answers
354 views

How to get a symmetric positive definite 5x5 matrix?

\begin{pmatrix} 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{pmatrix} is an example. But I don't find ...
0
votes
1answer
100 views

Which 6x6 line-matrix corresponds to a 4x4 point/plane-matrix

In 3-dimensional projective geometry I have a point-point map (collineation) $c$ with matrix $A$. Then $A^{-1t}$ is the matrix for the plane-plane map for the same $c$. These matrices are 4x4 and ...
1
vote
0answers
33 views

The group $O(n)$ is contained in a sphere of radius $n$.

So I have this exercise where I have to show that the group $O(n)$ is contained in a sphere of radius $n$ and centered on the origin, but I keep getting the wrong answer. Previously in the exercise I ...
1
vote
2answers
74 views

$A^m = r_m(A)?$ Power of a matrix!

In my Linear Algebra textbook we are reading, the following is stated for computing the power of a matrix in one of the advanced chapters as an exercise, $A^m = r_m(A)$. $r_m (A)$ is the remainder ...