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
6
votes
3 answers

Matrix Multiplication Understanding

I have to multiply two 3x3 matrices. I don't understand the answer. $$A=\begin{pmatrix}1&0&1\\ 0&1&1\\ 0&0&1 \end{pmatrix}$$ $$B=\begin{pmatrix}1&0&-1\\ 0&1&-1\\ 0&0&1 \end{pmatrix}$$ The answer given online is $$AB=\begin{pmatrix}1&0&1\\ 0&1&1\\…
Roy Sheehan
  • 359
6
votes
1 answer

Zeilberger's determinant evaluation problem

Doron Zeilberger posted a question here: http://arxiv.org/abs/1401.1532 of evaluating the determinant of a very sparse matrix. The $2d \times 2d$ matrix $M(d)$ has ones in the pattern $1,0,1,0,1,0,\dots$ in both the subdiagonals and the…
Craig Feinstein
  • 1,078
6
votes
3 answers

An old test question proving $\|\mathbf{B} - \mathbf{A}\| \lt \frac{1}{\|\mathbf{A}^{-1}\|}$ implies invertiblity of $\mathbf{B}$

I have an old test question that I am not sure about and would like some idea. It is from a Numerical Analysis class. Suppose that $A$ is an invertible $n$-by-$n$ matrix. Prove that for every $n$-by-$n$ matrix $B$, the inequality $$ \|\mathbf{B} -…
Shaza
  • 61
6
votes
2 answers

A matrix eigenvalue problem

In my previous problem, I made a typo. Now I restate it as a new problem. Let $ \begin{bmatrix} A& B \\ B^* &C \end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$. Is it true that $$\quad…
Sunni
  • 4,536
6
votes
2 answers

A property of positive definite matrices

Assume we have 2 positive definite matrices A and B . Show that there exists a non-singular matrix S such that - SAS' = I SBS' = L Here I is the Identity matrix and L is a diagonal matrix. S' is the transpose of S.
user92191
  • 61
6
votes
2 answers

Powers of matrices: if $A^3 = A$, how can we show that $A^{27}= A$?

Suppose that $A^3 = A$. How can we show that $A^{27}= A$? Any guidance would be much appreciated. Is it sufficient or correct to write that: $$A^9=(A^3)^3=(A)^3= A \implies A^{27}= (A^9)^3 = (A)^3=A ?$$
jaykirby
  • 852
6
votes
4 answers

How to calculate what matrix will transform specified points to other specified points

I want to transform an image. As far as I was able to find out, I can achieve this with a matrix, right? So here is my problem: how do I get this matrix if the only thing I know are the following starting and ending…
WiiMaxx
  • 163
6
votes
3 answers

How to establish that an anti-idempotent matrix is singular?

In linear algebra, idempotent matrices are defined by $$ A^2 = A \tag{1} $$ for a square matrix $A$. Obviously, the identity matrix $I$ is an idempotent matrix. It can be also shown that if $M$ is idempotent, then $I - M$ is idempotent by a trivial…
Dr. Sundar
  • 2,677
6
votes
1 answer

If $A^kX=B^kY$ for all $k$, is $X=Y$?

This one is more general than the one I asked before. Given invertible matrices $A,B$ and matrices $X,Y$ all with size $n$, such that $A^k X = B^k Y$ for $k=1,2,...,2n$. Does it follow that $X = Y$? I have no idea where to work. Thanks for any help.
user81767
  • 211
6
votes
4 answers

Every primitive matrix is irreducible?

$A$ is reducible if there is some permutation matrix $P$ such that $$ PAP^T = \begin{bmatrix} B & C \\ O & D \\ \end{bmatrix} $$ And, if $A^k > O$ for some k, then $A$ is called primitive. Then, how can I show that every primitive matrix is…
plhn
  • 631
6
votes
2 answers

Cute problem: determinant of $I_n+(f_if_j)_{i,j}$

I thought of the following little problem. Given numbers $f_1,\dots f_n$, what is the determinant of the symmetric matrix $I_n+(f_if_j)_{i,j}$? I have found a cute combinatorial-style proof that it is $1+\Sigma_i f_i^2$. using the sum over…
math_lover
  • 5,826
6
votes
6 answers

If $A^2 = I$, $B^2=I$ and $(AB)^2=I$, then $AB = BA$

Matrix Question If $A^2 = I$, $B^2=I$ and $(AB)^2=I$, then $AB = BA$ Basically, got up to $A(BA-AB)B = 0$ (by cancelling and equating terms from $I^2 = I$ and to $A^2B^2 = A^2B^2$ and using distributive laws), but that doesn't work out too…
user73229
  • 333
  • 3
  • 10
6
votes
1 answer

how to use matrix to prove this identity?

if $a_{n},b_{n}$ such $a_{0}=b_{0}=1$ $$\begin{cases}a_{n}=5a_{n-1}+7b_{n-1}\\ b_{n}=7a_{n-1}+10b_{n-1},\forall n=1,2,3,\cdots \end{cases}$$ show that $$a_{m+n}+b_{m+n}=a_{m}a_{n}+b_{m}b_{n}$$ It's an interesting identity, and I've proved it with…
math110
  • 93,304
6
votes
1 answer

Hadamard Product Matrix Norm Inequality

Let $A \odot B$ denote the Hadamard or entry-wise product of two matrices with equivalent dimensions. In this post ( Hadamard product: Optimal bound on operator norm ) it is claimed without proof that if $A$ is positive-definite, then $$\|A \odot…
BenB
  • 837
6
votes
2 answers

Why use a matrix?

Why would I want to use a matrix? It's good for organizing a few numbers but I can't find too much use for them. Could someone explain?
alexyorke
  • 420
1 2 3
…
99 100