Real Matrix Space

Why

We can add and scale matrices, so the $m \times n$ matrices are a vector space over $\R $.

Definition

The matrix sum of two matrices $A, B \in \R ^{m \times n}$ is the matrix $C \in \R ^{m \times n}$ defined by $C_{ij} = A_{ij} + B_{ij}$. In other words, the matrix $C$ is given by summing the entries of $A$ and $B$ “entry-wise”. We denote the matrix sum by $A + B$.

For $\alpha \in \R $, the $\alpha $-scaling of $A \in \R ^{m \times n}$ is the matrix $C \in \R ^{m \times n}$ defined by $C_{ij} = \alpha A_{ij}$. In other words, the matrix $C$ is given by scaling the entries of $A$ “entry-wise”. We denote the $\alpha $-scaled version of $A$ by $\alpha A$. These two definitions are justified by the following.

The matrix space (or matrix vector space) is the vector space $\R ^{m \times n}$ in which addition is given by the matrix sums and scalar multiplication by entry-wise scaling. Sometimes we reference explicitly the $m\times n$ matrix space.

Subspaces of symmetric matrices

Consider the space of square matrices $\R ^{n \times n}$. It is obvious that the subset of symmetric matrices is a subspace. Adding two symmetric matrices gives a symmetric matrix. Scaling a symmetric matrix by a real number gives a symmetric matrix. The matrix all of whose entries are 0 is symmetric.