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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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Why are these determinants $0$?

These $3$ matrices below have determinants of $0$. Increasing each element by $1$ still results in a determinant of…
Numb3r5
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Is a nonsingular matrix not the same as an invertible matrix?

Over an arbitrary ring $R$, a matrix $A$ is said to be invertible if it has an inverse with entries in the same ring. This happens iff $\det A$ is a unit of $R$. I've always thought that the terms "invertible" and "nonsingular" are synonymous. But I…
user557
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Are there matrices such that $(AB-BA)^{71}=I_{69}$?

Are there matrices $A,B \in \mathcal{M}_{69}(\mathbb{C})$ such that $$(AB-BA)^{71}=I_{69}?$$ Here $I_{69}$ denotes the $69 \times 69$ matrix with $1$ on its main diagonal and $0$ everywhere else. My strong guess is that there are not. Let…
AndrewC
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Possible Jordan Canonical Forms Given Minimal Polynomial

I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester. I took the polynomial to mean that there were at least two 2's and one 1 on the…
Frank White
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Finding $\mbox{tr}(A)$ from the condition $\mbox{tr}(A^2) = \mbox{tr}(A^3) = \mbox{tr}(A^4)$?

Let $A$ be an $n \times n$ matrix with real eigenvalues such that $$\mbox{tr}(A^2) = \mbox{tr}(A^3) = \mbox{tr}(A^4)$$ Then what would be $\mbox{tr}(A)$? I thought of finding $\sum_{i=1}^{n} \lambda_{i}$ from $$\sum_{i=1}^{n} \lambda_{i}^2 =…
BAYMAX
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Möbius function from random number sequence

Consider some arbitrary number sequence like the decimal expansion of $\pi$ = {3, 1, 4, 1, 5, 9, 2}. Prepend the sequence with the number $1$ so that you get {1, 3, 1, 4, 1, 5, 9, 2}. Then plug it into the first column in a matrix that has the…
Mats Granvik
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When does the adjacency or incidence matrix of a graph have consecutive ones property?

Given a graph, what are some sufficient (and necessary) conditions to tell if its adjacency matrix has the consecutive ones property? Similar question for its incidence matrix? Note that a $\{0,1\}$-valued matrix is said to have the consecutive ones…
Tim
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Determinant of a matrix after changes

If $\det \begin{pmatrix}a&1&d\\ b&1&e\\ c&1&f\end{pmatrix}=1$ and $\det \begin{pmatrix}a&1&d\\ b&2&e\\ c&3&f\end{pmatrix}=1$, what is $\det \begin{pmatrix}a&-4&d\\ b&-5&e\\ c&-6&f\end{pmatrix}$? So I am aware about all the different operations and…
sktsasus
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Powers of a simple matrix and Catalan numbers

Consider $m \times m$ anti-bidiagonal matrix $M$ defined as: $$M_{ij} = \begin{cases} -1, & i+j=m\\ \,\,\ 1, & i+j=m+1\\ \,\,\, 0, & \text{otherwise} \end{cases}$$ Let $S_n$ stand for the sum of all elements of the $n$-th power of the…
user
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Polynomial approximation to formal power series matrix

I noticed that starting with a $2 {\times} 2$ matrix $M$ with a handful of polynomial entries in two variables such that $\det M$ is invertible in $\mathbb C [x] [[y]]$ I can add infinitely many terms of higher orders in $y$ to the entries of $M$ to…
Earthliŋ
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Prove that $(AB-BA)^n=0$

Let $A,B$ be two $n \times n $ matrices with real entries and $\alpha \in \mathbb{R}, \alpha \neq 0$. It is given that $$A^2-B^2=\alpha(AB-BA)$$ Prove that $(AB-BA)^n=0$. At first, I tried to prove that $A(AB-BA)=(AB-BA)A$, as it would imply…
Shroud
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Probability that a random binary matrix is positive semi-definite

Let $A$ be a random $n \times n$ matrix such that $A_{ij}\in\{0,1\}$. Assume that each element $A_{ij}$ equals 1 with some probability $p>0$ and that all the draws are independent across elements. What is the probability that the symmetric part of…
user_lambda
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how does addition of identity matrix to a square matrix changes determinant?

Suppose there is $n \times n$ matrix $A$. If we form matrix $B = A+I$ where $I$ is $n \times n$ identity matrix, how does $|B|$ - determinant of $B$ - change compared to $|A|$? And what about the case where $B = A - I$?
DDR
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$\det(A^4+I)=29$ is not solvable by any $A\in M_4(\mathbb Z)$

I recently encountered the following problem. Given any $A \in M_4(\mathbb Z)$, show that $\det(A^4+I)\ne29$, where $I$ denotes the identity matrix. LHS can be written as the product of $1+{\lambda _i}^4$ where $\lambda _i$ denotes the eigenvalues…
kazuki
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How to prove that exponential kernel is positive definite?

The exponential kernel is defined by: $$k(x,z) = e^{-\alpha\|x-z\|}$$ where $\alpha>0$, $x,z\in \Bbb{R}^d$, $\|x\|$ is the 2-norm. The kernel matrix is defined by $K_{ij} = k(x_i,x_j)$, $i,j\in[1\ldots n]$. How to prove that $K$ is a positive…
itsuper7
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