Question

Show that if B is m × n, then BTB is positive semidefinite; and if B is n × n and invertible, then BTB is positive definite.

Answer 1

To show that BTB is positive semidefinite, we need to prove that x^TBTBx >= 0 for any nonzero vector x. If B is invertible, then BTB is positive definite.

Step-by-step explanation:

To show that BTB is positive semidefinite, we need to prove that xTBTBx ≥ 0 for any nonzero vector x. Let's start by expanding the expression:

xTBTBx = (Bx)T(Bx) = (Bx)T(Bx) = ||Bx||2

Since the norm of any vector is nonnegative, ||Bx||2 ≥ 0. Therefore, BTB is positive semidefinite.

If B is invertible, then BTB will be positive definite. This means that xTBTBx > 0 for any nonzero vector x. Since B is invertible, Bx = 0 only when x = 0. Therefore, ||Bx||2 > 0 for any nonzero vector x, and therefore BTB is positive definite.