**Chapter 6**

**Matrices and functions**

**6.0 Introduction**

If *A* is a matrix and *f*(·) is a function, what could be meant by *f*(*A*)? There are many interesting examples of functions *f: M*_{m,n} → *M*_{p,q}, depending on the values of *m, n, p*, and *q* and the nature of the functional relationship. Familiar examples of such functions are tr *A* and det *A* (if *m = n*), vec *A, A → AA*^{*} (*p = q = m*), *A → A*^{*} *A* (*p = q = n*), *A* → the *k*th compound of , *A* → *A* ⊗ *B* (*B* *M*_{r, s’} p = *mr, q* = *ns*), *A* → *f*(*A*) (*f*(·) a polynomial, *p = q = m = n*), *A → A* ∘ *A* (*p = m, q = n*).

If *m = n* = 1, we have ...