Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

A matrix is a rectangular array of elements, usually numbers or variables, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has $m \times n$ elements and is called an $m$ by $n$ matrix. Matrices are a part of .

Matrices can be added and subtracted. Furthermore, if they have compatible shapes, they can be multiplied. More precisely, given two matrices $A$ and $B$, the matrix $AB$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. In particular, given a natural number $n$, any two matrices $A$ and $B$ with $n$ columns and $n$ rows can be multiplied in both ways (that is, both $AB$ and $BA$ exist).

For questions specifically concerning matrix equations, use the tag.

55954 questions
3
votes
2 answers

Rotating a matrix itself, not applying a rotation to a space

How could I notate a matrix rotation? Example: $ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix},\:\:\: A_{\text{rotated}} = \begin{pmatrix} c & a \\ d & b\end{pmatrix}$. Notice the whole matrix is "rotated" clockwise. Is there any notation for…
ChemiCalChems
  • 366
3
votes
1 answer

Inverse of special type of symmetric block matrix

$W = \begin{pmatrix}A & B &B &\cdots&B\\ B& A & B &\cdots &B\\ \vdots & \vdots & \vdots & \ddots &\vdots &\\ B& B & B &\cdots &A \end{pmatrix} $ where $A$ and $B$ are symmetric matrices of appropriate order. In fact $B = aJ$, where $J$ is the matrix…
Satya Prakash
  • 367
3
votes
2 answers

Definition of Matrix Multiplication

I'm a linear algebra student and I've just come across the formal definition for multiplying matrices as follows: Let $A = \alpha_{ij}$ be an $l \times m$ matrix over $K$ and let $B = \beta_{ij}$ be an $m \times n$ matrix over $K$. The product $AB$…
mimyo
  • 198
  • 10
3
votes
0 answers

What Is This Matrix Operation Called And What Is It Used For?

I was over on stack overflow when I came across an algorithm based questions asking for help. I started working on it, but once I got the poster's example working the poster deleted the question. Thinking about it a little more, I wondered what this…
MrJman006
  • 131
  • 2
3
votes
0 answers

Show that every matrix can be in row-reduced echelon form

I would like to prove that every matrix can be in row-reduced echelon form. I started as follows: It's easy to show that any $2\times2$ matrix can be in row-reduced echelon form. Now assume that for $n\in N$, we have that $n-1\times n-1$ matrices…
Sha Vuklia
  • 3,960
  • 4
  • 19
  • 37
3
votes
2 answers

sum of two determinants

If \begin{vmatrix}a&c\\b&d\end{vmatrix} and \begin{vmatrix}a&e\\b&f\end{vmatrix}, then sum of these determinant can be written as in terms of another determinant given by \begin{vmatrix}a&c+e\\b&d+f\end{vmatrix} is it right?
Wajahat Ahmed
  • 47
3
votes
2 answers

Find matrices $A$ and $B$ given $AB$ and $BA$

Given that: $$AB= \left[ {\matrix{ 3 & 1 \cr 2 & 1 \cr } } \right]$$ and $$BA= \left[ {\matrix{ 5 & 3 \cr -2 & -1 \cr } } \right]$$ find $A$ and $B$.
Piseth
  • 135
3
votes
2 answers

Why do I get this matrix the wrong way round?

The question is: The set B = {${1+t^2, t+t^2, 1+2t+t^2}$} is a basis for P2. Find the coordinate vector of $p(t)=1+4t+7t^2$ relatvive to B. I made a matrix: \begin{bmatrix} 1 & 0 & 1 & 1\\ 0 & 1 & 2 & 4\\ 0 & 1 & 0 & 6 \end{bmatrix} When…
Rory McEwan
  • 579
3
votes
3 answers

Is every 1x1 matrix a diagonal matrix?

By definition of a diagonal matrix, a square matrix is said to be diagonal if all its non-diagonal elements are zero. So, can a 1x1 matrix be considered diagonal by this definition?
Zwolf
  • 145
  • 1
  • 5
3
votes
2 answers

What should be the value of a for it to be a singular matrix

for what value of a, $\begin{bmatrix}2a & -1\\-8 & 3\end{bmatrix}$ is a singular matrix. Can you also explain to me how to prove that a matrix is a singular matrix?
Dhruv Raghunath
  • 331
3
votes
1 answer

Is this matrix positive semi-definite?

$$A=\left[\frac{1}{i+j}\right]=\left(\begin{matrix}\frac{1}{1+1}&\frac{1}{1+2}&\cdots&\frac{1}{1+n}&\\\frac{1}{2+1}&\frac{1}{2+2}&\cdots&\frac{1}{2+n}\\\vdots&\vdots&\ddots&\vdots\\\frac{1}{n+1}&\frac{1}{n+2}&\cdots&\frac{1}{n+n}\end{matrix}\right)$$…
voltemp
  • 31
3
votes
1 answer

Why for any doubly stochastic matrix, there exist a permutation $\pi$, such that $A_{[i,\pi(i)]}\ne 0$?

Or put it simply, for any $n$-by-$n$ doubly stochastic matrix $A$, you can always find $n$ non-zero entries in $A$, none of them lies in the same row or column. Why is that?
xzhu
  • 4,193
3
votes
1 answer

Find $2 \times 2$ submatrix with sum greater than $3n$ in a $n \times n$ matrix whose sum is $n^3$

Let $n \ge 3$ be a positive integer. Inside a $n$ × $n$ array there are placed $n^2$ positive numbers with sum $n^3$. Prove that we can find a square $2$ × $2$ of $4$ elements of the array, having the sides parallel with the sides of the array,…
Macindows
  • 292
  • 1
  • 8
3
votes
1 answer

If matrix A commutes with B, and B commutes with C, and B is non invertible, is it true that A commutes with C?

Original question asked: If matrix $A$ commutes with $B$, and $B$ commutes with $C$, then does matrix $A$ commute with $C$? This can easily be disproven by taking $B = I$ and looking at some matrices $A$ and $C$ that doesn't satisfy the…
Squirrel-Power
  • 3,638
3
votes
1 answer

Rank of $n \times n$ matrix

Let $x,y \in K$ and $n \in \mathbb{N}$. Determine the rank of the following $n \times n$ Matrix: $xI_n$ + $y\cdot \sum_{i\neq j}E_{ij}$ in dependence of $x$ and $y$. $I_n$ is the identity matrix and $E_{ij}$ is the standard matrix. The matrix has…
MassU
  • 133
1 2 3
…
99 100