Matrices

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

This is essentially the process of verifying that two posts from [Linear Algebra], §Matrices and §Fundamental Theorem of Linear Algebra, remain valid for a general \(A\)-module. Most definitions and theorems are not difficult to state in this generality, but several complications arise from issues such as the possibility that \(A\) is not commutative, or that we will also treat infinite-dimensional free modules.

In this post we define matrices and the operations on them.

Basic Definitions of Matrices

As in [Linear Algebra], an \(m\times n\) matrix over a ring \(A\) is an arrangement of \(mn\) elements of \(A\) in an \(m\times n\) rectangular array

\[X=\begin{pmatrix}x_{11}&x_{12}&\cdots&x_{1n}\\x_{21}&x_{22}&\cdots&x_{2n}\\\vdots&\vdots&\ddots&\vdots\\x_{m1}&x_{m2}&\cdots&x_{mn}\end{pmatrix}\tag{1}\]

The set of all such matrices is denoted \(\Mat_{m\times n}(A)\). More generally, we define the following.

Definition 1 For arbitrary sets \(I, J\), an \(I\times J\) matrix is a family \((x_{ij})_{(i,j)\in I\times J}\) of elements of \(A\) indexed by \(I\times J\), and the collection of all such matrices is denoted \(\Mat_{I\times J}(A)\).

The difference from the usual definition is that \(I\) and \(J\) may be infinite. However, in order to define matrix multiplication and to examine the relationship between matrices and linear maps, these index sets must necessarily be finite, so the above definition is not a substantial generalization.

Definition 2 For an arbitrary \(I\times J\) matrix \(X\) and subsets \(I_0\subseteq I\), \(J_0\subseteq J\), the \(I_0\times J_0\) submatrix of \(X\) is defined as the matrix \((x_{ij})_{(i,j)\in I_0\times J_0}\) consisting of those \(x_{ij}\) with \(i\in I_0\) and \(j\in J_0\). In particular, the \(\{i\}\times J\) submatrix of \(X\) is called the \(i\)th row vector of \(X\), and the \(I\times \{j\}\) submatrix is called the \(j\)th column vector of \(X\).

We also define the following.

Definition 3 For an arbitrary \(I\times J\) matrix \(X=(x_{ij})_{(i,j)\in I\times J}\), the transpose \(X^t=(x^t_{ji})_{(j,i)\in J\times I}\) of \(X\) is the matrix given by the formula

\[x_{ji}^t=x_{ij}\qquad\text{for all $(i,j)\in I\times J$}\]

In particular, given an \(m\times n\) matrix \(X\) as in (1), its transpose is the \(n\times m\) matrix \(X^t\in\Mat_{n\times m}(A)\) given by

\[X^t=\begin{pmatrix}x_{11}&x_{21}&\cdots&x_{m1}\\ x_{12}&x_{22}&\cdots&x_{m2}\\\vdots&\vdots&\ddots&\vdots\\ x_{1n}&x_{2n}&\cdots&x_{mn}\end{pmatrix}\]

The following proposition is immediate from the definition.

Proposition 4 \((X^t)^t=X\).

Moreover, the \(i\)th row vector of \(X\) is the \(i\)th column vector of \(X^t\), and the \(j\)th column vector of \(X\) is the \(j\)th row vector of \(X^t\).

Operations on Matrices

The goal of this section is to show that \(\Mat_{I\times J}(A)\) carries the structure of an \(A\)-module.

First, we show that \(\Mat_{I\times J}(A)\) has the structure of an abelian group. For two matrices

\[X=(x_{ij})_{(i,j)\in I\times J},\qquad Y=(y_{ij})_{(i,j)\in I\times J}\]

in \(\Mat_{I\times J}(A)\), we define their sum by

\[X+Y=(x_{ij}+y_{ij})_{(i,j)\in I\times J}\]

Then by the associativity and commutativity of addition in \(A\), the above formula defines an abelian group structure on \(\Mat_{I\times J}(A)\). The identity element for addition in \(\Mat_{I\times J}(A)\) is the \(I\times J\) matrix all of whose entries are \(0\in A\), and the additive inverse of \(X\) is

\[-X=(-x_{ij})_{(i,j)\in I\times J}\]

We can then verify the identity

\[(X+Y)^t=X^t+Y^t\]

Now we must equip \(\Mat_{I\times J}(A)\) with a left \(A\)-action. For arbitrary \(\alpha\in A\) and \(X\in\Mat_{I\times J}(A)\) as above, we define

\[\alpha X=(\alpha x_{ij})_{(i,j)\in I\times J}\]

and the properties required for a left \(A\)-module follow from the fact that \(A\) is a left module over itself. Similarly, we define a right \(A\)-action on \(\Mat_{I\times J}(A)\) by

\[X\alpha=(x_{ij}\alpha)_{(i,j)\in I\times J}\]

We can verify that \((\alpha X)^t=\alpha X^t\) and \((X\alpha)^t=X^t\alpha\).

Moreover, \(\Mat_{I\times J}(A)\) is a free \(A\)-module, with basis given by the matrices

\[E_{ij}=(e_{kl})_{(k,l)\in I\times J},\qquad \text{$e_{kl}=0$ for all $(k,l)$ except for $e_{ij}=1$}\]

Strictly speaking, \(E_{ij}\) does not carry information about the size of the matrix, so this is a slight abuse of notation; however, in most cases there is no danger of confusion, so we use it as is. Then \((E_{ij})^t=E_{ji}\).

Matrix Multiplication

As in elementary linear algebra, for two matrices \(X\in\Mat_{I\times J}(A)\) and \(Y\in\Mat_{K\times L}(A)\), the product \(XY\) is defined only when \(J=K\), and in this case \(XY\) is an \(I\times L\) matrix. Following the entrywise expression for the matrix product examined after §Matrices, ⁋Definition 2, in the general setting the \((i,l)\)-entry of \(XY\) is defined by the formula

\[(XY)_{il}=\sum_{j\in J} X_{ij}Y_{jl}\tag{2}\]

However, since the right-hand side of the above formula need not be a finite sum in general, when we speak of matrix multiplication we always assume that \(J=K\) is a finite set.

Computing entrywise, we can verify the identities

\[(XY)Z=X(YZ),\quad X(Y+Z)=XY+XZ,\quad (X+Y)Z=XZ+YZ\]

However, unlike §Matrices, ⁋Proposition 8, there is a subtlety when considering the relationship between matrix multiplication and the transpose: the identity

\[(XY)^t=Y^tX^t\]

does not hold in general. This can be checked immediately by writing out both sides entrywise, and the reason is clear—namely, that \(A\) is not commutative. The above identity would make sense if we regarded both matrices \(X^t, Y^t\) on the right-hand side as matrices defined over the opposite ring of \(A\), but in most cases we will consider the case where \(A\) is commutative, so this is not of great significance.

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References

[Goc] M.S. Gockenbach, Finite-dimensional linear algebra, Discrete Mathematics and its applications, Taylor&Francis, 2011.
[Lee] 이인석, 선형대수와 군, 서울대학교 출판문화원, 2005.

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