Let A ∈ Mat n×n(R) be a real square matrix. (a) Suppose that A is symmetric,...

Let A ∈ Mat n×n(R) be a real square matrix.

(a) Suppose that A is symmetric, positive semi-definite, and orthogonal. Prove that A is the identity matrix.

(b) Suppose that A satisfies A = −A^T . Prove that if λ ∈ C is an eigenvalue of A, then λ¯ = −λ.

From now on, we assume that A is idempotent, i.e. A^2 = A.

(d) Set V1 = {v ∈ C n | Av = v} and V0 = {v ∈ C n | Av = 0}. Show that im A = V1 and ker A = V0.

(e) Prove that A is diagonalizable.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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