Definition 5.2.1. Similar Matrices.
A matrix \(A\) is similar to a matrix \(B\) if and only if \(A=PBP^{-1}\) for some invertible matrix \(P\text{.}\)
Section 5.2 Diagonalization
Goals.
We will
- define diagonalizable matrices,
- discover one use for these matrices,
- discover how to write a matrix in this format, and
- discover when they exist.
Subsection 5.2.1 Definitions
Definition 5.2.2. Diagonalizable.
A matrix \(A\) is diagonalizable if and only if it is similar to a diagonal matrix.
Subsection 5.2.2 Motivation
Remember, good mathematicians are lazy, we work very hard at finding easier ways of doing things. This includes wanting to reduce the number of arithmetic operations we have to peform for any calculation. The following is an example of reducing arithmetic operations.
Activity 5.2.1.
We will use the following matrices.
\begin{equation*}
P = \left[ \begin{array}{rr} -1 & 2 \\ 3 & 1 \end{array} \right].
\end{equation*}
\begin{equation*}
P^{-1} = \left[ \begin{array}{rr} -1/7 & 2/7 \\ 3/7 & 1/7 \end{array} \right].
\end{equation*}
\begin{equation*}
D = \left[ \begin{array}{rr} 10 & 0 \\ 0 & 3 \end{array} \right].
\end{equation*}
\begin{equation*}
A = \left[ \begin{array}{rr} 4 & -2 \\ -3 & 9 \end{array} \right].
\end{equation*}
Note \(A^2 = A A\text{,}\) that is matrix powers are repeated multiplication.
(a)
Calculate \(PDP^{-1}\text{.}\) Technology is great here.
(b)
Expand the expression \((PDP^{-1})^2\text{.}\) Your result will be an expression not a matrix with numbers.
(c)
Simplify this expression.
Hint.
Use associativity and look for things that can be reduced (go away).
(d)
Expand and simplify the expression \((PDP^{-1})^3\text{.}\)
(e)
Calculate \(D^2\text{.}\) Do by hand: you will see the pattern.
(f)
Calculate \(D^3\text{.}\) Do by hand: you will see the pattern.
(g)
Calculate \(D^{27}\text{.}\) Technology is not allowed here: use the pattern.
Hint.
Find a pattern from the previous cases.
(h)
If \(A=PDP^{-1}\) for some pair of matrices, how many matrix multiplication operations are required for \(A^{57}\text{?}\)
This activity indicates that in a special case powers of matrices are faster to calculate. This leads to the question, when can we write a matrix in this form?
Subsection 5.2.3 Method
Now that we have one reason to want diagonalizable matrices, we will look for a condition for when a matrix can be re-written this way. In the process we connect diagonalization with eigenvectors.
Activity 5.2.2.
Use the following matrices. \(A = \left[ \begin{array}{rr} 4 & -2 \\ -3 & 9 \end{array} \right]\) \(P = \left[ \begin{array}{rr} -1 & 2 \\ 3 & 1 \end{array} \right]\) \(D = \left[ \begin{array}{rr} 10 & 0 \\ 0 & 3 \end{array} \right]\text{.}\)
Note \(A\vec{x}_1=\lambda_1\vec{x}_1\) and \(A\vec{x}_2=\lambda_2\vec{x}_2\) where \(\vec{x}_i\) is the ith row of P and \(\lambda_1=10\) \(\lambda_2=3\text{.}\)
(a)
Calculate \(AP\text{.}\)
(b)
Calculate \(PD\text{.}\)
(c)
Compare \(AP\) to \(PD\text{.}\)
(d)
What are the columns of \(P\) with respect to the matrix \(A\text{?}\)
(e)
What are the non-zero entries of \(D\) with respect to the matrix \(A\text{?}\)
(f)
Based on what worked here, if a matrix is diagonalizable, what do \(P\) and \(D\) look like?
Finally we want to figure out when we can diagonalize a matrix (i.e., when will the process from above work).
Activity 5.2.3.
The purpose of this activity is to conjecture based on examples a condition that ensures we can diagonalize a matrix.
(a)
Using the process discovered above rewrite the following as \(A=PDP^{-1}\) when possible.
(i)
\(A=\left[\begin{array}{rrr} 4 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 1 \end{array} \right]\text{.}\) Eigenvalues are \(1,2,3\text{.}\)
(ii)
\(B=\left[\begin{array}{rrr} 1 & 3 & 3 \\ -3 & -5 & -3 \\ 3 & 3 & 1 \end{array} \right]\text{.}\) Eigenvalues are \(1,-2\text{.}\)
(iii)
\(C=\left[\begin{array}{rrr} 4 & 0 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{array} \right]\text{.}\) Eigenvalues are \(4,5\text{.}\)
(b)
Based on these examples, under what condition will a matrix be diagonalizable?
