Chapter 4, Problem 16SE

Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n.

16. If A is an m × n matrix of rank r, then a rank factorization of A is an equation of the form A = CR, where C is an m × r matrix of rank r and R is an r × n matrix of rank r. Such a factorization always exists (Exercise 38 in Section 4.6). Given any two m × n matrices A and B, use rank factorizations of A and B to prove that

rank (A + B) ≤ rank A + rank B

[Hint: Write A + B as the product of two partitioned matrices.]