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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.

55954 questions
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Name the matrix (not the game show)

I have a matrix of the following form: $ \begin{matrix} a_1 & 0 & \ldots & 0 \\ a_2 & a_1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ a_n & a_{n-1} & \ldots & a_1 \end{matrix} $ I'd like to know if this matrix has a special name (or…
Dany Zatuchna
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Proof - Square Matrix has maximal rank if and only if it is invertible

Could someone help me with the proof that a square matrix has maximal rank if and only if it is invertible? Thanks to everybody
markzzz
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Given that $A$ is a projection, $A$ is Hermitian if and only if $AA^\ast A=A$

In this question, $A^\ast$ is the conjugate transpose of $A$. I am asked to show that if $A$ is a projection matrix, that $A$ is Hermitian if and only if $A=AA^\ast A$. One direction is easy--if $A$ is Hermitian, the result is trivial. So what about…
Plutoro
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Jaccard index, matrix notation

I have a matrix with rows representing events and columns representing users. The elements of the matrix are binary values indicating if a user has attended the event or not. \begin{bmatrix}1&1&0&1&1\\1&1&0&0&1\\ 1&0&0&1&1\end{bmatrix} I need…
Morty
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$4 \times 4$ matrix and its inverse. Is my method ok?

I have been struggling to derive inverse matrix of a $4 \times 4$ Lorenz matrix $\Lambda$. $$ \Lambda = \begin{bmatrix} \gamma&0&0&-\beta \gamma \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ -\beta \gamma & 0 & 0 & \gamma \end{bmatrix} $$ My professor says…
71GA
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How to prove an equality

The matrices $A,B,C$ all $2\times 2$ dimensions so: $$A^2 +B^2 +C^2 =AB+BC+CA.$$ Prove that $$(A^2 +B^2 +C^2 -BA-CB-AC)^2=O_2$$ Can someone help me with this? Thank you.
AlexG
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If $A \in M_2(\mathbb R)$ non identical with $A^3=I $ then $\text{tr}(A)=-1$

Let $A \in M_2(\mathbb R)$ a $2\times 2$ matrix with real coefficient, such that $A \ne I$ and $$ A^3=I $$ Then $\text{tr}(A)=-1$. What if we consider $M_n(\mathbb R)$? Is the statement still true? I didn't manage to solve it, but I have a…
Romeo
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Attain matrix transpose by row swaps and column swaps

Given a matrix and the two operations row swap and column swap, how to prove the possibility or impossibility of getting the transpose via a composition of these operations?
math wannabe
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What is a necessary and sufficient condition of that $A$ has $n$ linearly independent eigenvectors?

If $A$ is an $n$-by-$n$ matrix with complex entries, (i.e., $A\in M_n(\mathbb{C})$,) $A$ must have $n$ eigenvalues, counting algebraic multiples. But it is not always true that $A$ has $n$ linearly independent eigenvectors. So, what necessary and…
xzhu
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When would you do matrix addition and multiplication to the same matrix?

Matrix addition, where we add corresponding elements to eachother, is much more similar to scalar addition, than matrix multiplication, where we add linear combinations of sub-vectors of the matrices, is to scalar multiplication. It seems to me that…
user56834
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Left and right multiplication of a matrix A with an upper triangular matrix U

Let A and $U$ are two square matrices where A is invertible. If $$AU=UA$$ In partitioned form…
Midhun Kathanaruparambil
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Nonsingular matrices with bounded coefficients

I can show that there exists $n^2$ positive integers $a_1,\ldots ,a_{n^2}$, such that each $n\times n$ matrix with coefficients $a_i$ (used once and only once) is nonsingular. Two questions: Could we find the smallest integer $M_n$ such that we can…
user350208
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General solution of a vectorial ecuation

This is the first time that I ask a question here. When I was looking for the maximum of a multivariable vector function, I encountered the following problem: I cannot find the general solution of the following vectorial equation: $$…
Luks
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Is it easier to determine that a matrix is singular than it is to determine nonsingular?

I came across this line "It is often easier to determine that a matrix is singular than it is to determine that a matrix is nonsingular. The facts below illustrate this. Fact 1.10. Let $A\in\mathbb C^{n\times n}$ and $x,b\in\mathbb C^n$. If…
Seetha Rama Raju Sanapala
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Rank of upper triangular matrix

Show that the rank of an upper triangular matrix is at least as large as the number of non-zero main diagonal entries. What I do not understand with this statement is how can one have a triangular matrix with more linearly independent vectors than…
crlb
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