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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.

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When will $AB=BA$?

Given two square matrices $A,B$ with same dimension, what conditions will lead to this result? Or what result will this condition lead to? I thought this is a quite simple question, but I can find little information about it. Thanks.
JSCB
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Matrices - Conditions for $AB+BA=0$

The Problem Let $A$ be the matrix $\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix} \bigr)$, where no one of $a,b,c,d$ is $0$. Let $B$ be a $2\times 2$ matrix such that $AB+BA=\bigl(\begin{smallmatrix} 0&0\\ 0&0 \end{smallmatrix} \bigr)$. Show…
Sarastro
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Is $AA^T$ positive semidefinite?

I have a very short question. Is $AA^T$ positive semidefinite, i.e. $x^T AA^Tx\geqslant 0$ for suitable $x$?
mathfemi
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Why is only a square matrix invertible?

Can anyone give a very simple proof (or explanation) as to why only square matrix can possibly be invertible?
Ylyk Coitus
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How to calculate the number of factorizations of a square matrix?

I need to write a function, that, given a square matrix M of non-negative integers, calculates the number of representations of M as a product of two square matrices of non-negative integers. Could you please help me with it?
Marty Colos
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Diagonalizable vs full rank vs nonsingular (square matrix)

There are many discussions of such type problems (comparison), for example: Diagonalizable vs Normal Today, I want to clearly understand the topic. Suppose the matrix $A\in \mathbb{R}^n$. Since the multiplication of all eigenvalues is equal to…
sleeve chen
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Does there exist a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$ such that $\mathbf{A}^{2}=-\mathbf{I}$?

Is it possible for a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$, $$\mathbf{A}^2=-\mathbf{I}$$ I know that It is possible for $2\times2$ matrix, but is it possible for $3\times3$ matrix ?
wantToLearn
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Additivity of the matrix exponential of infinite matrices

It is well known that the matrix exponential of finite dimensional matrices is additive if the exponents commute: $AB=BA\implies e^Ae^B=e^{A+B}$ (cf. e.g. Bernstein, Corollary 11.1.6). Under what circumstances does it carry over to the infinite…
Evan Aad
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Is there a way to extract the diagonal from a matrix with simple matrix operations

I have a square matrix A. Is there a way I can apply operations like addition, subtraction, matrix multiplication, matrix inverse and transpose to get the diagonal of the matrix. For example having: $$\begin{pmatrix}1&2\\3&4\end{pmatrix}$$ I would…
Salvador Dali
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What kind of matrix is it that when multiplied with its transpose produces the identity?

If $A^TA = I$, where $A$ is a lower triangular matrix, does that mean $A$ has to be an identity matrix (and nothing else)? In general, which kind of matrix $A$ must be for that equality to hold?
user1510863
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How to calculate the degrees of freedom of an $r$-ranked matrix with the size being $n\times n$?

Treat matrices as vectors lying in $\mathbb{R}^{n^2}$. It can be imagined matrices with rank $r (r
Ziyuan
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Prove $BA - A^2B^2 = I_n$.

I have a problem with this. Actually, still don't have the right way to start :/ Problem : Let $A$ and $B$ be $n \times n$ complex matrices such that $AB - B^2A^2 = I_n$. Prove that if $A^3 + B^3 = 0$, then $BA - A^2B^2 = I_n$. Thanks for any help.
user81767
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Prove $(A^T)^{-1}$ = $(A^{-1})^T$ whenever $A$ is invertible.

Prove $(A^T)^{-1}$ = $(A^{-1})^T$ for any invertible matrix $A.$ I actually don't know where to start. I do not think I can just apply index laws. Any help is cool! Thanks.
Clinton
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How to calculate matrix raised to a high power.

How would I go about calculating: $ \left( \begin{array}{cc} 3 & -\sqrt{3} \\ \sqrt{3} & -1 \end{array} \right)^{13}$ I have already attempted to find eigenvalues/eigenvectors, but I believe I am missing a much simpler method.
Sam
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Exponential lower bound for the determiant of a (0,1)-matrix

Give matrices, which only contain 0 and 1, and their determinant grows exponentially. In other words, show an $n \times n$ matrix for all n, which only contains 0 and 1, and $$\det A(n)>d \cdot c^n,$$ where c>1 and d>0. Can't really begin, any…
Atvin
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