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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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Matrix on complex field.

An $n\times n$ complex matrix $A$ satisfies $A^k=I_n$, where $I_n$ is $n\times n$ identity matrix and $k$ is positive integer $\gt1$. Suppose that 1 is not an eigen value of $A$. Then which of the fallowing is necessarily true? 1) $A$ is…
aryan
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Calculating the rank of a matrix , reduced row echelon or row echelon?

I am trying to calculate the rank of a matrix and everytime I search for the steps required to calculate the rank of a matrix, the answer always uses the terms row echelon form and reduced row echelon form interchangeably when calculating the rank…
Nubcake
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If a matrix as well as its Hermitian part both have determinant one, must the matrix be Hermitian?

If $x\in\mathrm{M}_2(\mathbb{C})$, $y=\dfrac{x+x^{\dagger}}{2}$, and $z=\dfrac{z-z^{\dagger}}{2}$, then $x=y+z$. Also, $y$ and $z$ respectively are Hermitian and anti-Hermitian, i.e. $y^{\dagger}=y$ and $z^{\dagger}=-z$, where $^\dagger$ denotes the…
j0equ1nn
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Can a constant be considered as 1x1 matrix?

A constant $c$ can be considered as 1 x 1 matrix $\;$ $( c )$ , it makes sense in terms of matrix inverse, matrix addition etc but the multiplication of a constant is possible to any matrix ( by multiplying all entries of matrix with it ) but if we…
advocateofnone
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Matrix norm question

Let $A^*$ denote the complex conjugate transpose of a matrix $A$. In the Euclidean norm, if $$||A^*A+AA^*||=||A^*A||$$ does it imply that $AA^*=0$. If not, could you give a counter-example?
PeterA
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How can we show that $(I-A)$ is invertible?

$A$ is an $n\times n$ matrix with $\|A\|≤a<1 $ . I need to prove that the matrix $(I-A)$ is invertible with $\|(I-A)^{-1}\|\le\frac1{(1-a)}$. It doesn't say anything more. The norm makes me confuse. How can we start to solve this. Could you please…
rose
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Assume that $A $ is an $n \times n$ symmetric positive-definite matrix.

Assume that $A$ is an $n\times n$ symmetric positive-definite matrix. Prove that: the element of $A$ with maximum magnitude must lie on the diagonal.
sami
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If $A=\pmatrix{1 &0\\-1&1}$, show that $A^2-2A+I_2=0$. Hence find $A^{50}$

If $$A=\pmatrix{1 &0\\-1&1},$$ show that $$A^2-2A+I_2=0,$$ where $I_{2}$ is the $2x2$ Identity matrix. Hence find $A^{50}$. We have $$A^2-2A+I_2=A(A-2I_3)+I_=\pmatrix{1 &0\\-1&1}\pmatrix{-1 &0\\-1&-1}+I_2 =-I_2+I_2=0.$$ How can I show the…
user181545
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Inverse of a sum of symmetric matrices

How can I proof this result? $$ X(X+Y)^{-1}Y=(X^{-1} +Y^{-1})^{-1} $$ where $X$, $ Y$, $(X+Y)$, and $(X^{-1} +Y^{-1})$ are symmetric and invertible matrices,each one with dimensions $p\times p$.
Stefano R.
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Maximal commutative subring of the ring of $2 \times 2$ matrices over the reals

Motivated by complex numbers, I noticed that the set of all elements of the following forms a commutative sub-ring of $M_2(\mathbb{R})$: \begin{pmatrix} x & y\\ -y & x \end{pmatrix} Is this sub-ring maximal w.r.t commutativity? If this sub-ring is…
Isomorphism
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Solving a linear equation for a symmetric,positive matrix

Given the Problem $A x = b$ for some regular matrix $A \in \mathbb{R}^{n \times n}$ and $b\in\mathbb{R}^n$. One can compute $x$ with the Cholesky factorization in $O(n^3)$. If $A$ is known to be a symmetric, positive (i.e. $A_{ij} >0$) and positive…
Adam
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When does a strictly diagonally dominant matrix have dominant principal minors?

$A$ is an $N\times N$ matrix with diagonal elements $a_{ii}=1-s_{i}$, and off diagonal elements $a_{ij}=s_{i}w_{ij}$ for $i≠j$. Assume $0≤s_{i}<1/2$ and $\sum_{j≠i}w_{ij}=1$ for all $i$ and $0≤w_{ij}≤1$. As $1-s_{i}>∑_{j≠i}a_{ij}=s_{i}∑_{j≠i}w_{ij}$…
M Engineer
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Rotation matrices are similar if and only if their angles add up to 2 pi

Let $\theta_0, \theta_1 \in [0, 2\pi)$ and $\theta_0 \ne \theta_1$. Consider the rotation matrices $$M_0 = \left[ \begin{matrix}\cos(\theta_0) & -\sin(\theta_0) \\ \sin(\theta_0) & \cos(\theta_0) \\ \end{matrix} \right],M_1 = \left[…
rehband
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Matrix properties

What can one conclude about a matrix, $M$, if its single eigenvalue is 1? (I think the question is trying to demonstrate a contrast with the case where it is 0 instead of 1, in which we could conclude that the matrix is nilpotent.) Can I conclude…
impotent
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Is the square root of a symmetric positive definite matrix also symmetric?

The inverse of a SPD matrix is also symmetric. But what about the square root? Intuitively, I would say yes. But I'm not sure about it.
colorblind
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