91Ó°ÊÓ

Writing Can a matrix be similar to two different diagonal matrices? Explain.

Proof Prove that if matrix \(A\) is diagonalizable, then \(A^{T}\) is diagonalizable.

Find the matrix \(A\) of the quadratic form associated with the equation. Then find the eigenvalues of \(A\) and an orthogonal matrix \(P\) such that \(P^{T} A P\) is diagonal. $$17 x^{2}+32 x y-7 y^{2}-75=0$$

In Exercises \(43-52,\) find a matrix \(P\) such that \(P^{T} A P\) orthogonally diagonalizes \(A\) Verify that \(P^{T} A P\) gives the correct diagonal form. $$A=\left[\begin{array}{ll} 4 & 2 \\ 2 & 4 \end{array}\right]$$

Guided Proof Prove that nonzero nilpotent matrices are not diagonalizable. Getting Started: From Exercise 80 in Section 7.1 you know that 0 is the only eigenvalue of the nilpotent matrix \(A\). Show that it is impossible for \(A\) to be diagonalizable. (i) Assume \(A\) is diagonalizable, so there exists an invertible matrix \(P\) such that \(P^{-1} A P=D,\) where \(D\) is the zero matrix. (ii) Find \(A\) in terms of \(P, P^{-1},\) and \(D\) (iii) Find a contradiction and conclude that nonzero nilpotent matrices are not diagonalizable.

In Exercises \(43-52,\) find a matrix \(P\) such that \(P^{T} A P\) orthogonally diagonalizes \(A\) Verify that \(P^{T} A P\) gives the correct diagonal form. $$A=\left[\begin{array}{rrr} 0 & 10 & 10 \\ 10 & 5 & 0 \\ 10 & 0 & -5 \end{array}\right]$$

Proof Prove that if \(A\) is a nonsingular diagonalizable matrix, then \(A^{-1}\) is also diagonalizable.

Cayley-Hamilton Theorem In Exercises \(49-52\) demonstrate the Cayley-Hamilton Theorem for the matrix \(A\). The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of \(A=\left[\begin{array}{rr}1 & -3 \\ 2 & 5\end{array}\right]\) is \(\lambda^{2}-6 \lambda+11=0,\) and by the theorem you have \(A^{2}-6 A+11 I_{2}=O\). $$A=\left[\begin{array}{rr}5 & 0 \\\\-7 & 3\end{array}\right]$$

Use the Principal Axes Theorem to perform a rotation of axes to eliminate the \(x y\) -term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. $$2 x^{2}+4 x y+2 y^{2}+6 \sqrt{2} x+2 \sqrt{2} y+4=0$$

Prove that if a symmetric matrix \(A\) has only one eigenvalue \(\lambda,\) then \(A=\lambda I\)