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Question: 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 rankr, then a rank factorization of A is an equation of the form A=CR, where C is an m×r matrix of rankr 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)rankA+rankB.

(Hint: Write A+B as the product of two partitioned matrices.)

Short Answer

Expert verified

It is proved that rank(A+B)rankA+rankB.

Step by step solution

01

Show that rank(A+B)rankA+rankB

Consider rankA=r1 and rankB=r2. Rank factorization of A and B are A=C1R1 and B=C2R2, where C1 is m×r1 with rankr1, C2 is m×r2 with rankr2, R1 is r1×n with rankr1, R2 is r2×n with rankr2.

By stacking R1 over R2, create a m×(r1+r2) matrix C=(C1C2) and a (r1+r2)×n matrix R.

A+B=C1R1+C2R2=(C1C2)(R1R2)=CR

The matrix CR is a product, so according to exercise 12, its rank cannot exceed the rank of either of its factors. The rank of C cannot exceed r1+r2 because C has r1+r2. Similarly, as matrix R has r1+r2 rows, then its rank cannot exceed r1+r2.

Therefore, the rank of A+B cannot exceed r1+r2=rankA+rankB, or rank(A+B)rankA+rankB.

Thus, it is proved that rank(A+B)rankA+rankB.

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