Prefactory Comments on Matrices and Vectors

A special case of matrix multiplication occurs when we deal with matrices having all nonzero elements in square blocks along the diagonal, such as the following two  

The most conspicuous feature of this product matrix is that it is blocked out in exactly the same way as are its factors. It is not difficult to see that this sort of result must always be obtained. Moreover, it should also easily be seen that the elements of a given block in the product matrix are determined only by the elements in the corresponding blocks in the factors. Thus, when two matrices which are blocked out along the diagonal in the same way are to be multiplied, the corresponding blocks in each may be considered independently of the remaining blocks in each. Specifically, in the above case, 

An important property of a square matrix is its character. This is simply the sum of its diagonal elements, and it is usually given the symbol % (Greek chi). Thus 

We shall now prove two important theorems concerning the behavior of characters. 

Conjugate matrices are related by a similarity transformation in the same way as are conjugate elements of a group. Thus, if matrices //l and. are conjugate, there is some other matrix J such that 

---