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
32 views

Why left multiplication when it comes to Markov chains?

When working with Markov chains and transition matrices $P$ we multiply from the left, meaning that for example $\mu^{(n)} = \mu^{(0)}P^n$ or that the stationary distribution satisfies $\pi = \pi P$. ...
0
votes
1answer
33 views

What is the minimum number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the minimum number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My ...
2
votes
3answers
46 views

How to reverse matrix vector multiplication?

I'm using the simple matrix x vector multiplication below to calculate result. And now I wonder how can I calculate ...
1
vote
3answers
47 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
0
votes
2answers
37 views

$A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, show that $\text{rank}(AB)\le\text{rank}(A)$.

The problem is asking a proof for $\text{rank}(AB)$ is smaller or equal to $\text{rank}(A)$. Given the conditions $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Any idea about the ...
1
vote
1answer
23 views

matrix function onto and 1-1

I have just started a linear algebra paper and we are doing 1-1 and onto functions. I understand in theory what they mean, I just don't know how to prove them. For example: Define $f: ...
1
vote
1answer
20 views

Multiplication of diagonal matrices with identity

What would the result of this multiplication be, given that $A$ is an $m \times n$ rectangular diagonal matrix and $I$ is the identity matrix. $$A^TIA = \cdots$$
2
votes
1answer
55 views

Invertibility of $I-AB$ [duplicate]

I got a question in linear algebra: 1) Let A and B be $n\times n$ matrices. If $I - AB$ is an invertible matrix, then prove that $I - BA$ is invertible. Can someone tell me how to solve this ...
0
votes
1answer
31 views

How to write this system in the form Ax=b

Given the following system of N equations with N unknowns, with $\lambda$ known and the $a_{ij}$'s also known entries of an m*n matrix A. How would you express the system in the form A x=b? x is of ...
2
votes
2answers
109 views

Prove that if the sum of each row of A equals s, then s is an eigenvalue of A. [duplicate]

Consider an $n \times n$ matrix $A$ with the property that the row sums all equal the same number $s$. Show that $s$ is an eigenvalue of $A$. [Hint: Find an eigenvector] My attempt: By definition: ...
1
vote
0answers
28 views

Iterative methods for solving a linear equation system

There are several methods known for solving a linear equation system Ax = b (like Jacobi or Gauss-Seidel) by iterating $x_{n+1}=Mx_n+c$ with a matrix M, for which some norm is smaller than 1. But ...
4
votes
0answers
66 views
+50

solving equation also involving unknown matrix in trace

Given two real $m$ x $k$ matrices $A_1$ and $B_1$ and two $k$ x $k$ real matrices $A_2$ and $B_2$ I want to solve the following equation for $Q$. $Q$ is an orthogonal matrix, i.e. $Q^TQ=I$. ...
0
votes
0answers
6 views

Transform gradient to reference element

Minimal example of the problem My attempt I think this is not a linear solution like \begin{equation} \nabla u = \nabla A_K x + \nabla b_K \end{equation} which must be wrong because $A_K$ is a ...
2
votes
1answer
49 views

Basis for space of matrices in $\mathbb M_2(\mathbb R)$

Given that $G=\left\{ \left(\begin{array}{cc} a & -a\\ b & c \end{array}\right):a,b,c\in\mathbb{R}\right\} $ and $H=\left\{ \left(\begin{array}{cc} x & y\\ z & -z ...
2
votes
1answer
30 views

A generalization of GMRES

In oder to solve $Ax=b$, GMRES method finds $x_n$ in the $k$-th Krylov subspace i.e.: $$K_n=span\{b,Ab,...,A^{n-1}\}$$ and we have the condition: minimize $\|r_n\|_2$, which $r_n=b-Ax_n$ Now we ...
2
votes
3answers
318 views

Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
-1
votes
0answers
24 views

Calculate distance between two objects based on their visible height for a specific focal length

How do I calculate the distance between to objects of the same size base on their height for a given focal length. Both object 1 and object 2 are 15 cm in height (actual size). Object 2 looks ...
1
vote
1answer
20 views

Automatic defining global variables in MATLAB

How could I define a set of variables automatically in MATLAB. For example: kt1 to kt_n*(2*n+1) ...
1
vote
1answer
35 views

How to factor and reduce a huge determinant to simpler form? Linear Algebra

So, I have learned about cofactor expansion. But the cofactor expansion I know doesn't reduce the number of rows and colums to one matrix. I usually pick a colum, multiply each element in the column ...
0
votes
0answers
43 views

A problem on matrices over a commutative ring

Let $M_{m,n}(R)$ denote an $m\times n$ matrix with each entry over a commutative ring $R$, $m\leq 2\leq n$, and there is a matrix $\mathbf{B} = M_{m,n}(R)$. $\mathbf{B}\mathbf{s} = \mathbf{a}$, where ...
2
votes
3answers
67 views

Real matrices with non-real eigenvalues

I know this covers a lot, so perhaps someone could redirect me to a helpful website. for a) I have no idea where to start on the proof, as I don't understand why this is true. for b) I also have ...
2
votes
1answer
25 views

Column space of stochastic matrix.

Consider an arbitrary matrix $M \in \mathbb{R}^{n \times m}$. Denote the column space of $M$ as $\mathcal{C}(M)$. Is it always possible to construct a right stochastic matrix $S$ such that ...
0
votes
0answers
5 views

Notation for concatenation of indexed vectors/matrices

Is there any standard notation for concatenation of matrices/vectors where their indices are taken from a set. I have matrices $A_{ij}$ where $(i,j)\in S$. I want to denote a matrix $A$ which is the ...
1
vote
0answers
58 views

Integer matrices whose $m$-th power are identity matrix

How can one find all the matrices with integer entries of size $n \times n$ such that $A^{m}=I$ where $m$ is fixed integer and the matrix does not have fixed point in $\mathbb{Z}^n$ (except zero of ...
1
vote
1answer
59 views

Trace of power of stochastic matrix

I would like to know if this statement is true. Having a stochastic matrix (rows sum up to 1), with a positive (non-negative) diagonal, then it holds that $$\text{trace}({W^2})\leq ...
3
votes
1answer
48 views

$3 \times 3$ real matrix: relation with determinants

$A$ is a $3 \times 3$ matrix with real entries such that $\operatorname{det}(A+I_3)=\operatorname{det}(A+2I_3)$. Then is $2\operatorname{det}(A+I_3)+\operatorname{det}(A-I_3)+ 6 =3 ...
0
votes
1answer
31 views

Linear Equation as matrix

Using a series of 3x3 matrices multiplied together, it is possible to create a matrix which will rotate, translate, scale and invert a size 2 vector. Using a 4x4, it is possible to do this to a size ...
2
votes
1answer
23 views

The probability of getting a certain image by random pixelation

Well, seeing that I'm terribly bad at math I don't know how to solve this, I'll try to explain, excuse me if I sound dumb. Just suppose that I've got a photo/image with 320x240 resolution and 24 bit ...
1
vote
0answers
18 views

About reduction to Hessenberg matrix

I've read somewhere that Hessenberg decomposition is not unique unless the first column of $Q$ is given. i.e $Q^TAQ=H$ Then I read the algorithm of Arnoldi iteration and I found an amazing fact: ...
0
votes
0answers
16 views

Matrix multiplier for ODE

I have matrix C with dimensions $3 \times 3 $ and it is skew symmetric too C is given by $C(0,0)=0,C(1,1)=0,C(2,2)=0 \tag 1$ $C(1,0)= sc_0+ px (c_1-c_0),C(0,1)=-C(1,0) \tag 2 $ $C(0,2)= ...
1
vote
0answers
24 views

Solving tridiagonal matrices where the top left element is zero

If I have a matrix like this: $$ \left[\begin{array}{rrrrrrrrr|r} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & ...
1
vote
1answer
28 views

Express summation in terms of matrix norm

Express the following $$\sum _{ i=1 }^{ n }{ ({ \beta }_{ 1 }x_{ i }+{ \beta }_{ 0 }-y_{ i })^{ 2 } }$$ To become something of the form: $∥Ax−b∥^{ 2 }$ where $A$ is an $m$−by−$n$ matrix and $b$ is ...
0
votes
1answer
28 views

Matrix transpose times itself

We define A to be a matrix in $R^{m*n}$ Does $A^TA$ have any particular structure? When is $A^TA$ invertible?
1
vote
1answer
31 views

Find the Jacobian of F

Given that $A \in \mathbb{R}^{m\times n}$, and $b \in \mathbb{R}^{m}$, we define: $$F:\mathbb{R}^{n} \rightarrow \mathbb{R} = \left\| Ax-b \right\|^2$$ Find the Jacobian of $F$, and show that it is of ...
1
vote
1answer
27 views

Reduced Row Echelon form without scalar multiplication?

Is it possible to transform any matrix to row reduced echelon form without using the row operation that multiplies a row by a scalar?
0
votes
1answer
14 views

Determine if matrix D belongs to Vect(A,B,C)

So there are 4 matrices, A, B,C,D. They belong to field F5. Determine if D belongs to Vect(A,B,C). I have pretty much done all the calculations its just i fail to conclude/find the right value for the ...
4
votes
1answer
58 views

What is the name of the matrix that is created by a vector times its transpose.

I am looking for the name of the matrix created by the following operation: $Z = z*z^T$ I know it should create a symmetric matrix with an element $Z_{ij} = z_{i}z_{j}$
2
votes
0answers
39 views

Probability that a random integer matrix is singular

Let A be a nxn-matrix with integers in the range $u..v$ , where $u<v$ are arbitary integers. Is there a formula, or at least, a good estimate, for the probability that the matrix is singular ? ...
0
votes
0answers
14 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
4
votes
0answers
78 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
1
vote
2answers
40 views

Matrix Semi-Definite Inequality [duplicate]

Does the following inequality hold? If matrix $A$ is a $n \times n $ positive semi-definite, $A \succeq 0$, and $U$ is one $n \times k$ unit column-orthogonal matrix ($k \leq n$), $U^{T}U=I$, do we ...
0
votes
3answers
26 views

Generating a Triangular Matrix via a Vector MATLAB

How do I generate an arbitrary (size n) triangular matrix with a vector? For example: A=[1;2;3;4;5;6;7;8;9;10]; And the answer should be: B=[1,2,3,4; 0,5,6,7; 0,0,8,9; 0,0,0,10] or ...
2
votes
2answers
32 views

diagonalization of a matrix over finite fields

I'm having a problem with determine whether a matrix is diagonalizable over $\mathbb F_{2}$, over $\mathbb F_{3}$, etc. for example, for the following matrix: $$ \begin{bmatrix} 1 ...
1
vote
1answer
58 views

To find the volume of the region that is bordered by 4 points in 3D space

To find the volume of the region that in the points $A(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3),D(x_0,y_0,z_0)$. Let's define a 4X4 matrix to determine plane equation that are on $A,B,C$ ...
0
votes
2answers
54 views

string function

Jane loves string more than anything. She made a function related to the string some days ago and forgot about it. She is now confused about calculating the value of this function. She has a string ...
0
votes
0answers
18 views

Heaviside Expansion Theorem with matrices

Is the Heaviside Expansion Theorem (HE) for the determination of inverse Laplace Transforms true for matrix expressions such that $\mathscr{L}^{-1}[\mathbf{P}(s)\mathbf{Q}^{-1}(s)] = \sum_i^n ...
0
votes
3answers
25 views

How do you know that rows are independent and what are the 120 terms?

I am having trouble with the question below, help me out;
0
votes
0answers
27 views

Eigenvalue bounds for a positive semidefinite matrix

I have a symmetric $(p\times p)$, positive semi definite matrix $\Omega$. If somebody says: find the eigenvalue bounds of the matrix such that $$w_1I \le \Omega \le w_2I$$ where $I$ is the identity ...
2
votes
0answers
42 views

Top bound on the value of an algebraic adjunct to elements of a nonnegative irreducible matrix

Let $A = ||a_{i j}||_1^n$ be nonnegative irreducible matrix with maximum eigenvalue $r$. Let $A_{i j}(\lambda)$ be an algebraic adjunct for the element $\lambda \delta_{i j} - a_{i j}$ in determinant ...
0
votes
2answers
55 views

Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

What determinant is zero? What equation does this give for the plane? I need some help here, am pretty stuck