Math 303: Section 18

Dr. Janssen

\[ \def\R{{\mathbb R}} \def\b{{\mathbf{b}}} \def\x{{\mathbf{x}}} \def\v{{\mathbf{v}}} \def\w{{\mathbf{w}}} \DeclareMathOperator{\null}{Nul} \]

Computing powers of a matrix is often desirable.

Let \(D = \left[\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}\right]\).

Explain in general why

\[ D^n = \left[\begin{matrix} 2^n & 0 \\ 0 & 3^n \end{matrix}\right]. \]

This will be true in general!

Let \(D\) be any matrix, \(P\) an invertible matrix, and \(A = P^{-1} D P\).

- Show that \(A^2 = P^{-1} D^2 P\).
- Show that \(A^3 = P^{-1} D^3 P\).
- Explain in general why \(A^n = P^{-1} D^n P\).

**Definition 1 **The \(n\times n\) matrix \(A\) is *similar* to the \(n\times n\) matrix \(B\) if there is an invertible matrix \(P\) such that \(A = P^{-1} B P\).

Let \(A= \left[\begin{matrix} 1 & 1 \\ 2 & 0 \end{matrix}\right]\) and \(B = \left[\begin{matrix} 2 & 2 \\ 0 & -1 \end{matrix}\right]\).

- Let \(P = \left[\begin{matrix} 1 & -1/2 \\ -1 & 1 \end{matrix}\right]\); then \(P^{-1} = \left[\begin{matrix} 2 & 1 \\ 2 & 2 \end{matrix}\right]\). Verify that \(A = P^{-1} B P\) (perhaps with
`sage`

). - Calculate \(\det(A)\) and \(\det(B)\), and record your observations.
- Find the characteristic polynomials of \(A\) and \(B\). What do you notice?
- What can you say about the eigenvalues of \(A\) and \(B\)? Why?
- Explain why \(\x = \left[\begin{matrix} 1 \\ 1 \end{matrix}\right]\) is an eigenvector for \(A\) with eigenvalue 2. Is \(\x\) an eigenvector for \(B\) with eigenvalue 2? Why or why not?

- Similar matrices share some, but not all, properties
- Why don’t similar matrices have to have the same eigenvectors?

Let \(A\) and \(B\) be similar matrices with \(A = P^{-1} B P\).

- Use the multiplicative property of the determinant to explain why \(\det(A) = \det(B)\).
- Use the fact that \(P^{-1} I P = I\) to show that \(A - \lambda I\) is similar to \(B - \lambda I\).
- Explain why it follows from the previous two parts that

\[ \det (A - \lambda I) = \det(B -\lambda I). \]

**Theorem 1 (Properties of Similar Matrices) **Let \(A\) and \(B\) be similar \(n\times n\) matrices and \(I\) the \(n\times n\) identity matrix. Then:

- \(\det(A) = \det(B)\)
- \(A-\lambda I\) is similar to \(B - \lambda I\)
- \(A\) and \(B\) have the same characteristic polynomial
- \(A\) and \(B\) have the same eigenvalues.

A *matrix transformation* via an \(m\times n\) matrix \(A\) takes \(\R^n\) to \(\R^m\) via left multiplication.

Another use of similar matrices is to rewrite a given matrix transformation (with respect to the standard basis) using a “nicer” matrix.

Let \(A = \left[\begin{matrix} 7 & 2 \\ -4 & 1 \end{matrix}\right]\) define a transformation \(T: \R^2 \to \R^2\). We can verify that the eigenvalues of \(A\) are \(5\) and \(3\), corresponding to eigenvectors \(\v_1 = \left[\begin{matrix} 1 \\ -1 \end{matrix}\right]\) and \(\v_2 = \left[\begin{matrix} 1 \\ -2 \end{matrix}\right]\). If we let \(P = [ \v_1 \ \v_2]\) and \(D = \left[\begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]\), then

\[ A = P D P^{-1}. \]

**Definition 2 **Let \(A\) be \(n\times n\). We say \(A\) is *diagonalizable* if there is an invertible \(n\times n\) matrix \(P\) such that \(P^{-1} A P\) is a diagonal matrix.

- Let \(A\) be \(n\times n\).
- Find the eigenvalues of \(A\), and the corresponding eigenvectors.
- If there are \(n\) linearly independent eigenvectors \(\v_1, \ldots, \v_n\), then \(P = [ \v_1 \cdots \v_n]\), and \(D\) is formed by putting the eigenvalues on the main diagonal
*in order*, repeating as necessary.**Congratulations! \(A\) is diagonalizable**. - If there are
*not*\(n\) linearly independent eigenvectors, then \(A\) is not diagonalizable.

Find, if possible, an invertible matrix \(P\) that diagonalizes \(A\), and write \(A = P D P^{-1}\).

- \(A = \left[\begin{matrix} 1 & 1 \\ 0 & 2 \end{matrix}\right]\)
- \(A = \left[\begin{matrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{matrix}\right]\)
- \(A = \left[\begin{matrix} 2 & 4 & 3 \\ -4 & -6 & -3 \\ 3 & 3 & 1 \end{matrix}\right]\)