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Section 7.2 Diagonalization
Definition 7.2.1.
A square matrix \(A\) is said to be diagonalizable if there exists an invertible matrix \(P\) and a diagonal matrix \(D\) such that
\begin{equation*}
P^{-1}AP = D
\end{equation*}
or equivalently,
\begin{equation*}
A = PDP^{-1}
\end{equation*}
The columns of
\(P\) are eigenvectors of
\(A\text{,}\) and the diagonal entries of
\(D\) are the corresponding eigenvalues of
\(A\text{.}\)
-
A
\(n\times n\) matrix
\(A\) is diagonalizable if and only if
\(A\) has
\(n\) linearly independent eigenvectors.
-
A
\(n\times n\) matrix
\(A\) is diagonalizable if and only if the geometric multiplicity equals the algebraic multiplicity for each eigenvalue.
-
If
\(A\) has
\(n\) distinct eigenvalues, then
\(A\) is diagonalizable.
Let \(A\) be a square matrix which is diagonalizable. Here is the command to find the matrix \(P\) and \(D\) such that
\begin{equation*}
P^{-1}AP=D.
\end{equation*}
Let check if the columns of
\(P\) are eigenvectors.
Exercise: For the matrix
\begin{equation*}
B = \begin{pmatrix}
1 \amp 2 \amp -2 \\
-2 \amp 5 \amp -2 \\
-6 \amp 6 \amp -3
\end{pmatrix}\text{,}
\end{equation*}
find the matrix \(P\) and \(D\) such that
\begin{equation*}
P^{-1}AP=D.
\end{equation*}
Exercise: If matrix
\(A\) has eigenvalues
\(1, 2, 3\) with corresponding eigenvectors
\(\begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}, \begin{bmatrix}1 \\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix}\text{,}\) find the matrix
\(A\text{.}\)
