Today we continue Section 4.3. Read 4.3 and Appendix C for next class. Work through suggested exercises.
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Question: If $P$ is invertible, how do $\det A$ and $\det(P^{-1}AP)$ compare?
Definition: If $A$ is $n \times n$, $\det (A - \lambda I)$ will be a degree $n$ polynomial in $\lambda$. It is called the characteristic polynomial of $A$, and $\det (A - \lambda I) = 0$ is called the characteristic equation.
By the fundamental theorem of invertible matrices, the solutions to the characteristic equation are exactly the eigenvalues.
1. Compute the characteristic polynomial $\det(A - \lambda I)$.
2. Find the eigenvalues of $A$ by solving the characteristic equation
$\det(A - \lambda I) = 0$.
3. For each eigenvalue $\lambda$, find a basis for the eigenspace
$E_\lambda = \null (A - \lambda I)$
by solving the system $(A - \lambda I) \vx = \vec 0$.
Theorem D.4 (The Fundamental Theorem of Algebra): A polynomial of degree $n$ has at most $n$ distinct roots.
Therefore:
Theorem: An $n \times n$ matrix $A$ has at most $n$ distinct eigenvalues.
Also:
Theorem D.2 (The Factor Theorem): Let $f$ be a polynomial and let $a$ be a constant. Then $a$ is a zero of $f(x)$ (i.e. $f(a) = 0$) if and only if $x - a$ is a factor of $f(x)$ (i.e. $f(x) = (x - a) g(x)$ for some polynomial $g$).
Example 4.18: Find the eigenvalues and eigenspaces of $A = \bmat{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4 \emat$.
Solution: 1. We computed the characteristic polynomial as $$ \det (A - \lambda I) = - \lambda^3 + 4 \lambda^2 - 5 \lambda + 2 $$ 2. By trying some small integers, we found that this factors as $$ \kern-4ex \det (A - \lambda I) = - (\lambda - 1)^2 (\lambda - 2) $$ with roots $\lambda = 1$ and $\lambda = 2$.
3. We found that the $\lambda = 1$ eigenspace $E_1$ has basis $\colll 1 1 1$. See text for $E_2$.
A root $a$ of a polynomial $f$ implies that $f(x) = (x-a) g(x)$. Sometimes, $a$ is also a root of $g(x)$, as we found above. Then $f(x) = (x-a)^2 h(x)$. The largest $k$ such that $(x-a)^k$ is a factor of $f$ is called the multiplicity of the root $a$ in $f$.
In the case of an eigenvalue, we call its multiplicity in the characteristic polynomial the algebraic multiplicity of this eigenvalue.
We also define the geometric multiplicity of an eigenvalue $\lambda$ to be the dimension of the corresponding eigenspace.
In the previous example, $\lambda = 1$ has algebraic multiplicity $\query{2}$ and geometric multiplicity $\query{1.}$ Also, $\lambda = 2$ has algebraic multiplicity $\query{1}$ (and the text shows that it has geometric multiplicity 1).
Example 4.19: Find the eigenvalues and eigenspaces of $A = \bmat{rrr} -1 & 0 & 1 \\ 3 & 0 & -3 \\ 1 & 0 & -1 \emat^\strut$. Do partially, on board.
In this case, we find that $\lambda = 0$ has algebraic multiplicity 2 and geometric multiplicity 2.
These multiplicities will be important in Section 4.4.
Theorem 4.15: The eigenvalues of a triangular matrix are the entries on its main diagonal (repeated according to their algebraic multiplicity).
Example: If $A = \bmat{rrr} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 1 \emat$, then $$ \kern-6ex \det(A - \lambda I) = \bdmat{ccc} 1-\lambda & 0 & 0 \\ 2 & 3-\lambda & 0 \\ 4 & 5 & 1-\lambda \edmat = (1 - \lambda)^2 (3 - \lambda) , $$ so the eigenvalues are $\lambda = 1$ (with algebraic multiplicity 2) and $\lambda = 3$ (with algebraic multiplicity 1).
Question: What are the eigenvalues of a diagonal matrix?
Question: What are the eigenvalues of $\bmat{cc} 0 & 4 \\ 1 & 0 \emat$?
Question: How can we tell whether a matrix $A$ is invertible using eigenvalues?
So we can extend the fundamental theorem with two new entries:
Theorem 4.17:
Let $A$ be an $n \times n$ matrix. The following are equivalent:
a. $A$ is invertible.
b. $A \vx = \vb$ has a unique solution for every $\vb \in \R^n$.
c. $A \vx = \vec 0$ has only the trivial (zero) solution.
d. The reduced row echelon form of $A$ is $I_n$.
f. $\rank(A) = n$
g. $\nullity(A) = 0$
h. The columns of $A$ are linearly independent.
i. The columns of $A$ span $\R^n$.
j. The columns of $A$ are a basis for $\R^n$.
k. The rows of $A$ are linearly independent.
l. The rows of $A$ span $\R^n$.
m. The rows of $A$ are a basis for $\R^n$.
n. $\det A \neq 0$
o. $0$ is not an eigenvalue of $A$
Theorem 4.18: If $\vx$ is an eigenvector of $A$ with eigenvalue $\lambda$, then $\vx$ is an eigenvector of $A^k$ with eigenvalue $\lambda^k$. This holds for each integer $k \geq 0$, and also for $k < 0$ if $A$ is invertible.
In contrast to some other recent results, this one is very useful computationally:
Example 4.21: Compute $\bmat{rr} 0 & 1 \\ 2 & 1 \emat^{10} \coll 5 1$.
Solution: By finding the eigenspaces of the matrix, we can show that $$ \kern-6ex \bmat{rr} 0 & 1 \\ 2 & 1 \emat \coll 1 {-1} = - \coll 1 {-1} \qtext{and} \bmat{rr} 0 & 1 \\ 2 & 1 \emat \coll 1 2 = 2 \coll 1 2 $$ Write $A = \bmat{rr} 0 & 1 \\ 2 & 1 \emat$, $\vx = \coll 5 1$, $\vv_1 = \coll 1 {-1}$ and $\vv_2 = \coll 1 2$. Since $\vx = 3 \vv_1 + 2 \vv_2$ we have $$ \begin{aligned} A^{10} \vx \ &= A^{10} (3 \vv_1 + 2 \vv_2) = 3 A^{10} \vv_1 + 2 A^{10} \vv_2 \\ &= 3 (-1)^{10} \vv_1 + 2(2^{10}) \vv_2 = \coll {3+2^{11}}{-3+2^{12}} \end{aligned} $$ Much faster than repeated matrix multiplication, especially if $10$ is replaced with $100$.
This raises an interesting question. In the example, the eigenvectors were a basis for $\R^2$, so we could use this method to compute $A^k \vx$ for any $\vx$. However, last class we saw a $3 \times 3$ matrix with two one-dimensional eigenspaces, so the eigenvectors didn't span $\R^3$. We will study this further in Section 4.4, but right now we can answer a related question about linear independence.
Theorem: If $\vv_1, \vv_2, \ldots, \vv_m$ are eigenvectors of $A$ corresponding to distinct eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_m$, then $\vv_1, \vv_2, \ldots, \vv_m$ are linearly independent.
Proof in case $m = 2$: If $\vv_1$ and $\vv_2$ are linearly dependent, then $\vv_1 = c \vv_2$ for some $c$. Therefore $$ A \vv_1 = A \, c \vv_2 = c A \vv_2 $$ so $$ \lambda_1 \vv_1 = c \lambda_2 \vv_2 = \lambda_2 \vv_1 $$ Since $\vv_1 \neq \vec 0$, this forces $\lambda_1 = \lambda_2$, a contradiction.$\quad\Box$
The general case is very similar; see text.
Next: how to become a Billionaire using the material from this course.