5.5 Matrix algebra

So far we have looked at matrices and their relationships to linear maps. We now investigate operations that are relevant when two or more matrices are involved. We cover the general case first of matrices which may have different numbers of rows and columns and then move on to square matrices.

All matrices are over fields in this section and when we are manipulating multiple matrices they all have entries in the same field. We can consider matrices over rings such as the integers but we have no need to make this restriction for quantum computing.

5.5.1 Arithmetic of general matrices

Matrices that are the same size, meaning they have the same number of rows and columns, can be added together entry by entry. For example,

The same is true for subtraction and negation.

Scalar multiplication is done entry by entry.

The set of n by m matrices over a field F for given integers n and m ≥...