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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1
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1answer
25 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?
1
vote
0answers
46 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
24
votes
1answer
762 views

Prove the determinant of this matrix

We have a square matrix, that all elements on main diagonal are zero, and other elements are following: $$a_{i,j}=\begin{cases} 1,&\text{if i+j belongs to Fibonacci numbers,}\\ 0,&\text{if ...
7
votes
1answer
56 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
0answers
88 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
votes
0answers
25 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
votes
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 $w_i \neq z_j$ ...
0
votes
2answers
40 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 ...
1
vote
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 ...
1
vote
2answers
52 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
2
votes
2answers
25 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 ...
4
votes
1answer
145 views

What is the connection between $\rho$ and $\sigma$ if $\rho\rho^T=\sigma\sigma^T$?

I want to prove that there exists a Borel function $R(\rho,\sigma)$ with values in $M^{d\times d}$ defined on $D=\lbrace(\rho,\sigma)\in M^{d\times d}\times M^{d\times d}\,: ...
0
votes
0answers
40 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
3answers
118 views

Do these matrices have any name?

Assume $A$ is a square matrix defined as follow: $$A=\sum_{i} u_{i}u_{i}^T$$ where for each $i$, $u_i$ is a non-negative column vector. Do the matrices of these forms have any special name?
0
votes
0answers
33 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\\ ...
3
votes
2answers
44 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
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 ...
1
vote
1answer
57 views
+50

Do these two rearranged matrices have the same singular values (or the same rank)?

This is the origin of my problem: I have a set of data which expresses which user ($U$ set) applies what tag ($T$ set) to which item ($I$ set). So it is actually a $U×I×T$ tensor $A$ (or 3-dimensional ...
1
vote
1answer
124 views

Tree Traversal - Simple Puzzle type Issue.

This is a puzzle like question,based on Fibonacci like structure of the tree. Actually it is a short question with out any complex concepts. It appears bit big,since I have added explanations with ...
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 ...
2
votes
1answer
65 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?
0
votes
2answers
38 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 ...
2
votes
1answer
286 views

Column Space and SVD

I was reading Gilbert Strang's book and he says that if $A=USV'$ be the SVD of A ( assume square for the moment) then the nullspace of A is given by the last $n-r$ columns of V and the column space by ...
3
votes
1answer
45 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 ...
3
votes
2answers
723 views

Matrix multiplication - Express a column as a linear combination

Let $A = \begin{bmatrix} 3 & -2 & 7\\ 6 & 5 & 4\\ 0 & 4 & 9 \end{bmatrix} $ and $B = \begin{bmatrix} 6 & -2 & 4\\ 0 & 1 & 3\\ 7 & 7 & 5 ...
2
votes
3answers
183 views

Basic question: what does the notation $[A,B]$ mean?

If $A$ and $B$ are both matrices, what is $[A,B]$? I understand that it is a commutator and that $[A,B]=AB-BA$, but since I don't know what a commutator is, none of this information is telling me ...
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 ...
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 ...
2
votes
1answer
587 views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
7
votes
4answers
111 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 ...
5
votes
3answers
118 views

Let $A$ be an $n\times n$ invertible complex matrix such that $A^7 = A^*$. Show that $A^8 = I$.

Let $A$ be an $n\times n$ invertible complex matrix such that $A^7 = A^*$ (where $*$ denotes conjugate transpose). Show that $A^8 = I$. Here are my thoughts so far: I was able to show that all ...
1
vote
2answers
42 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 ...
0
votes
1answer
14 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 ...
-1
votes
2answers
114 views

three questions on analytic geometry and matrices [on hold]

the lines $x-2y=4$ and $6x+ay=8$ are perpendicular. Calculate the value of $a$. prove that the matrix $\pmatrix{\cos\theta& \sin\theta\\ -\sin\theta & \cos\theta}$ ...
5
votes
1answer
70 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
0
votes
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 ...
17
votes
2answers
177 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 ...
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 ...
11
votes
3answers
152 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 ...
0
votes
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 ...
1
vote
1answer
39 views

Why are points from this matrix geometric sequence co-planar?

Let $ M= \left[ {\begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \\ \end{array} } \right] $, such ...
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 ...
0
votes
0answers
19 views

Positive definiteness and eigenvalue inequalities of two symmetric matrices

Let $A = A^T \in \mathbb{R}^{n \times n}$ and $B = B^T \in \mathbb{R}^{n \times n}$ be any symmetric matrices. Then, I know that if $A$ and $B$ are positive definite, and $A-B$ is positive ...
2
votes
1answer
41 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 ...
0
votes
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 ...
0
votes
1answer
13 views

Characteristic polynomial and characteristic equation

What is the major difference between the characteristic polynomial and the characteristic equation?
3
votes
4answers
27k views

Online tool for diagonalizing matrices?

I need some online tool for diagonalizing 2x2 matrices or at least finding the eigenvectors and eigenvalues of it. I don't like to download any stuf because I'm not able to, some online tool will do ...
2
votes
1answer
119 views

How to solve this $2\times2$ linear system of equations?

at the moment I am a little bit confused. Here is the matrix I am trying to solve $$ \left( \begin{array}{cc|c} 5 & -1& 12 \\ -1 & 2& 12 \end{array} \right) $$ I tried ...
1
vote
2answers
51 views

Finding the amount of solutions in a 3 equation solution

So, I'm not really sure how to calculate the amount of solutions for a system with 3 equations. All I know is that it has something to do with matrices and the discriminant, but I'm not sure where to ...
6
votes
2answers
3k views

The determinant of block triangular matrix as product of determinants of diagonal blocks

I am given the following partitioned - upper-triangular matrix: $$ \begin{bmatrix} A_1 &* &* &* &* &* \\ 0& A_2 &* &* &* &* \\ .& 0& ...