📂Linear Algebra

Matrix Spaces

Definitions

For a field $F$, let us define the set of $m \times n$ matrices whose components are elements of $F$ as $M_{m \times n}(F)$.

$$M_{m \times n}(F) := \left\{ \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix} : a_{ij} \in F \right\}$$

Then, with respect to matrix addition and scalar multiplication, $M_{m \times n}(F)$ is a $F$-vector space.

Description

The set of matrices of the same size is itself a vector space. This might seem obvious because if you arrange numbers in a line, you get a tuple (vector), and if you arrange them in a rectangle, you get a matrix.

Subspaces

Zero Trace Matrices

A matrix whose trace is $0$ is called a zero trace matrix.

The set of all $n \times n$ zero trace matrices $W$ is a $n^{2}-1$-dimensional subspace of $M_{n \times n}$. It’s easy to see that the dimension of $W$ is $n^{2}-1$. For example, considering the case when $3 \times 3$, $W$ is as follows:

$$W = \left\{ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & -(a+e) \end{bmatrix} \right\}$$

Therefore, $W$ is generated by the following set, and its dimension is $3^{2}-1 = 8$.

$$\beta = \left\{ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},\right. \\[1em] \qquad \qquad \left. \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \right\}$$

Upper Triangular Matrices

Let’s call the set of all $n \times n$ upper triangular matrices as $W$. Then, $W$ is a $\sum\limits_{k=1}^{n}k$-dimensional subspace of $M_{n \times n}$. For example, considering the case when $3 \times 3$,

$$W = \left\{ \begin{bmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{bmatrix} \right\}$$

The set generating this is as follows, and the dimension of $W$ is $1 + 2 + 3 = 6$.

$$\beta = \left\{ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \right. \\[1em] \qquad \qquad \left. \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \right\}$$