(See the discussion on p. 167 and Exercise 3.4.25.) Let A be an n n matrix. Prove that

Chapter 4, Problem 24

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QUESTION:

(See the discussion on p. 167 and Exercise 3.4.25.) Let A be an n n matrix. Prove that the functions A : R(A) C(A) and AT : C(A) R(A) are inverse functions if and only if A = QP, where P is a projection matrix and Q is orthogonal.

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QUESTION:

(See the discussion on p. 167 and Exercise 3.4.25.) Let A be an n n matrix. Prove that the functions A : R(A) C(A) and AT : C(A) R(A) are inverse functions if and only if A = QP, where P is a projection matrix and Q is orthogonal.

ANSWER:

Step 1 of 3

It is given that  is   matrix.

It is known that,  is a one-one function.

Suppose  and  are inverse functions.

Then,

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