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  3. let a ? mat n×n(r) be a real square matrix. (a) s

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

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Advance Math

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

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

(b) Suppose that A satisfies A = ?A^T . Prove that if ? ? C isan eigenvalue of A, then ?¯ = ??.

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

(c) Prove that if ? is an eigenvalue of A, then ? is equal to 0or 1.

(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.

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