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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4 views

distribution of a quadratic form

Let $A$ be a symmetric positive definite matrix, and $x$ a random vector. Suppose we know the distribution of $x^\top A x$. What can we say about the distribution of $x^\top x$?
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0answers
4 views

similarities between two binary matrices

I want to measure the similarities between two matrices A and B. Both A and B contains the feature vectors of sounds and are in binary format. i want to see what is the similarities between these two ...
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1answer
15 views

Working with zeroes in Gaussian elimination

I am trying to find the null space of a matrix mod 2. So far I have tried to implement basic Gaussian elimination. Something happened that should've been very easy to solve but it's late and I can't ...
1
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0answers
13 views

Convergence of a Sequence of Projection Matrices

Suppose I have a sequence of growing matrices $A_n$, and $B_n$, both of the same size, and both rows and columns are growing at the same rate for each step $n$. Furthermore, we assume that there ...
2
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0answers
17 views

When does a strictly diagonally dominant matrix have dominant principal minors?

$A$ is an $N\times N$ matrix with diagonal elements $a_{ii}=1-s_{i}$, and off diagonal elements $a_{ij}=s_{i}w_{ij}$ for $i≠j$. Assume $0≤s_{i}<1/2$ and $\sum_{j≠i}w_{ij}=1$ for all $i$ and ...
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0answers
29 views

Rank of the product of 3 matrices

Suppose I have 3 n by n matrices $A,B,C$ with $ABC=0$, what could be the maximal rank of $CBA$? I guess the answer would be n but I failed to prove it( tried to use Rank-Nuillity Theorem but I don't ...
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1answer
16 views

Is this symmetric, block-diagonal matrix positive semi-definite?

I have a matrix of the following form, where $a,b,c>0$ \begin{align*} A = \left[ \begin{array}{cccccc} aM_{12}^2 & aM_{12}M_{13} & 0 & 0 & 0 & 0 & 0 \\ aM_{13}M_{12} ...
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1answer
22 views

Inverse Matrix Multiplication

Let $A \in F^{n*n}$ a inverse matrix and $B\in F^{n*n}$ a none inverse matrix We can say that because A is row equivilate to $I_n$$ \implies $ $AB$ is none inverse matrix?
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0answers
27 views

Sum/diff of matrix units

I understand what the product of matrix units means, but I don't understand what the sum/difference of two different matrix units represents. For example, what does ${e_{2,2}}-{e_{5,5}} $ equal? ...
3
votes
1answer
40 views

Special solutions to Ax = 0

I solved most of it, just not sure about one point. The problem statement, all given variables and data Suppose A is the matrix shown below: $$ \begin{pmatrix} 0 & 1 & 2 ...
1
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1answer
41 views

A subset that is closed under multiplication but not addition? [duplicate]

I can't get my head around subspaces despite having studied on them quite a lot. Here goes: The problem statement, all given variables and data Give an example of a non-empty subset U of R^2 such ...
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0answers
41 views

Matrices over field with characteristic $p$

For $A,B$ $n\times n $ matrices over a field $F$ with characteristic $p$ if $AB-BA=cI$ for $c\in F$ does this imply that $c=0$? Intuitively I would say that it doesn't but I cannot think of a ...
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0answers
4 views

Dominance Network Worded Problems

What are some methods to solve this? Normally for dominance I do as such: Write a matrix for one step dominance, then find total dominance by = D+D^2 - then sum each row of the matrix. Using this ...
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0answers
13 views

Relationship between eigen-vector and adjacency matrix nodes

My question is short and simple. I am wondering the following: lets say I have a adjacency matrix of a graph lets say NxN and λ stands for the highest eigen-valueand u for the correspondant ...
0
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0answers
6 views

Principal submatrix rank property

how to prove that for positive semi definite matrices (nXn) , the rank fo a principal submatrix equals the rank of the expanded submatrix with the same rows but with all the original matrix columns? ...
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0answers
35 views

How can I calculate this matrix differentiation? [on hold]

I am studying about the Matrix Differentiation. I don't know if this red box differential metric, which is how it is calculated.
0
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0answers
28 views

Isomorphic matrix groups over rings

I've thinking about this problem for the last couple days and I can't get anywhere. I would really appreciate some help. Is it true that, a) $\operatorname{SL}_n(\mathbb{Z}/2013\mathbb{Z})\cong ...
0
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0answers
7 views

Convex combination of polynomials with roots on the unit circle and companion matrix

Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ ...
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2answers
42 views

When does the equality hold for norm equivalence

We know that for a vector $x\in \mathbb{R}^n$, its 1-norm and 2-norm satisfy that $$\frac{1}{n}\|x\|_1\le\|x\|_2\le \|x\|_1,$$ could anyone please give me some hints that on what condition these ...
2
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1answer
33 views

Metric on the Set of Binary rectangular matrices

Consider a set of all possible Binary rectangular matrices. How many non-equivalent metrics can be defined? How to define non equivalent metrics on this set precisely?
2
votes
3answers
41 views

Determinant-like expression for non-square matrices

I'm interested in whether for any real matrix of size $m \times n$ there is a real number with the following properties: It is a polynomial expression with real coefficients in the entries of the ...
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2answers
28 views

orthogonal matrices vs. orthogonal columns

I'm just reading a book on econometrics and now I'm stuck with a problem: There is a Theorem on "Orthogonal Partitioned Regression" which says: "In the multiple linear least squares regression of ...
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0answers
34 views

Solve linear equations [on hold]

\begin{bmatrix} 0 & 0& 1& 1& 1&0 \\ 0 & 0& 0& 0& 2&1 \\ -3 & 0 & 0 & 0 & -2& 0\\ 4& 4& 0& 0& 0 & -1\\ ...
2
votes
1answer
71 views

Why we use $\mathbb{R}^{m \times n}$ notation instead of $\mathbb{R}^{n \times m}$?

I just realised, that I use all the time the notation $\mathbb{R}^{n \times m}$, and all books and papers use $\mathbb{R}^{m \times n}$. $\mathbb{R}^{n \times m}$ is more sympathetic for me, because I ...
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3answers
52 views

Is the basis of null space of a matrix always a subset of the basis of its column space?

Given an $m\times n$ matrix $A$, is the basis of its null space (set of $x$ such that $Ax=0$) always a subset of the basis of the row space of $A$? In general, the basis of a subspace may not be a ...
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2answers
47 views

finding the closest matrix of a given form

let's say I have a vector $(a_1\dots a_n)$, where each component is between $-1$ and $1$. Now from this vector I define a $n\times n$ matrix $M$ such that $$M_{ij} = \begin{cases} 1&\,& i = ...
2
votes
1answer
66 views

Which two matrices will create the zero matrix multiplication

I was thinking, which property of matrices could help me determine if the multiplication of some $A$ and $B$ result the zero matrix?
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2answers
57 views

Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$

Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$. I appreciate your hints, Thanks
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2answers
45 views

Adding a constant to a matrix

Find p(A) if p(x) = $2x^2 - x + 1$ where A is the below matrix: $$ \begin{bmatrix} 3 & 1 \\ 2 & 1 \\ \end{bmatrix} $$ Attempt at a solution p(A) = $2 \cdot ...
3
votes
1answer
46 views

Is $\mathbf {B^TAB}$ non-singular for a non-singular $\mathbf A$, and $\mathbf {B}$ with full column-rank?

If $\mathbf A$ is any square non-singular matrix of dimension $n \times n$. And $\mathbf B$ is a $n \times m$ matrix with $\mathrm{rank(\mathbf B)} = m$. Is the full rank condition of matrix $\mathbf ...
0
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0answers
42 views

Follow-up on solution to markov process equation

I asked a question here about solving a system related to an absorbing markov chain. I now have a variation where there are $m$ types (of student, job seeker, etc) each of which applies to ...
2
votes
1answer
44 views

Calculating an integral with a matrix

I want to calculate the following integral: Let A be a symmetric, invertible matrix. $\int_{K}<A^2x,x>dx$ where $K:=\{x\in \mathbb R^n : \|Ax\|_2\leq1\}$ A is symmetric, hence there is an ...
7
votes
1answer
69 views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
4
votes
1answer
51 views

Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $

I'm having a hard time counting. I need to count the number of elements for the multiplicative group of invertible $2\times 2$ matrices $GL_{2}\left( \mathbb{F}_{p}\right) $ with elements from the ...
0
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1answer
16 views

Basic Matrix Properties

I know its basic but I am not quite getting it. I have two matrices W and U. W has 3M rows and M columns while U is M into M diagonal matrix. I want to ask if R1 and R2 are equivalent. If yes then ...
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2answers
43 views

How to show the identity relating to Matrix

Suppose that $$ A=\begin{bmatrix}a_{11}&a_{21}\\a_{21}&a_{22}\end{bmatrix}, \ \ B=\begin{bmatrix}d&-1\\1&0\end{bmatrix}. $$ and $$A=B^N$$ Show that $$a_{11}=\sum_{i=0}^{[N/2]}(-1)^i ...
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0answers
31 views

Transpose/multiplication of 3D matrices

I have $A(p)=\begin{bmatrix}p_1 &p_2 & p_3\\ 2p_1 &2p_2^2 & 4p_3^3\\ 3p_1 &3p_1 & 10\\ \end{bmatrix}\tag 1$ $ p= {\left(\begin{array}{c}p_1\\p_2\\p_3\\p_4 ...
0
votes
1answer
42 views

Unique least square solutions

There is a theorem in my book that states: If $A$ is $m\times n$, then the equation $Ax = b$ has a unique least square solution for each $b$ in $\mathbb{R}^m$. But can we find a counter-example to ...
0
votes
3answers
34 views

Is This A Image Of A linear Transformation?

Let there be $T:R^3 \rightarrow R^3$ $T(0,-1,1)=(3,3,3)$ $T(1,0,-1)=(0,1,1)$ $T(1,1,0)=(1,2,-1)$ Is (1,2,3) is the only image of the vector $(1, \frac{-7}{9}, \frac{-8}{9})$? I have thought to ...
5
votes
1answer
121 views
+50

Eigenvalue test faster than $O\left(n^3\right)$?

Given a real $n\times n$ matrix $A$, one can find the eigenvalues in $O\left(n^3\right)$ by using say, the $QR$ algorithm. Now, what if we guess an eigenvalue $\lambda_0$, and we want to know if it's ...
0
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0answers
12 views

Definite integral - Finding an equivalent form

I have the following definite integral $ \int_{0}^{L} {\psi(t) }_{1 \times 5}{A(s)}_{5 \times 5}(\psi(t) _{1 \times 5})^{T} {B(s)}_{5 \times 5} ds \tag 1 $ Given data All dimensions are ...
7
votes
4answers
114 views

How to find x so that $\|A x\| = \|A\| \|x\|$ holds

The subbordinance property of matrix-vector multiplication states that $\|A x\| \le \|A\| \|x\|$ where $\|x\|$ is the norm of vector $x$ and $\|A\|$ is the induced norm of matrix $A$. Many textbooks ...
2
votes
1answer
43 views

Complex matrix and diagonalizablity

Let $A\in\mathcal{M}_4(\mathbb C)$ such that $\operatorname{rank}(A)=2$ and $A^{3}=A^2$ $\neq0$. Suppose that $A$ is not diagonalizable. Then 1. One of the Jordan blocks of the Jordan cannonical form ...
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1answer
13 views

Characteristic polynomial and characteristic equation

What is the major difference between the characteristic polynomial and the characteristic equation?
0
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1answer
27 views

Transformation of inverse to a system of linear equations

I have $X = (U'WU)^{-1}U'$ to be solved. Suppose $U'$ is $3 \times 7, W$ is $7 \times 7$ positive definite matrix, $U'$ is of rank 3. So, I transformed $(U'WU)^{-1}U'$ as $(U'WU)^{-1}U'WU = I\\ XWU ...
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1answer
19 views

Open subset of the space of matrices

This question comes from the process of my learning about Grassmann manifolds. Suppose that $M(m,n)$ is the set of real $n \times m$ matrices, where $n>m$. Let $F(m,n)$ be a subset of $M(m,n)$ ...
0
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1answer
17 views

Similar vs congruent matrices

Suppose that some symmetric matrix $S$ (everything here is over the field of real numbers) is similar to a diagonal matrix $D$ via the invertible matrix $P$. We have: $P^{-1}DP=S.$ My question: ...
1
vote
1answer
25 views

Each element of a real orthogonal matrix is equal to its cofactor

If $A =(a_{ij})$ be a real orthogonal matrix with $\det A = 1$, prove that each element $a_{rs}$ of $A$ is equal to its cofactor $A_{rs}$ in $\det A$. I got this basic problem from my text book and ...
17
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2answers
179 views

Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.

I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n ...
11
votes
3answers
153 views

Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its ...