Let ( A ) have eigenvalues ( λ₁, ldots, λₙ ), and suppose that ( P ) diagonalizes ( A ). Show that for any positive integer ( k ), ( P ) diagonalizes ( Aᵏ ) and determine ( P⁻¹ A P ).
A) To confirm diagonalization for ( Aᵏ ), ( P ) must be an orthogonal matrix.
B) Diagonalization for ( Aᵏ ) holds irrespective of the diagonalizability of ( A ).
C) ( P ) diagonalizes ( Aᵏ ) if and only if ( A ) is a symmetric matrix.
D) ( P⁻¹ A P ) simplifies to ( diag(λ₁ᵏ, ldots, λₙᵏ) ).
