Announcements:

Today we (mostly) finish 4.3. Read Appendix C and section 4.4 for next class. Work through recommended homework questions.

Tutorials: Quiz 8 covers 4.2 and 4.3, including the parts of Appendix D that we covered.

Help Centers: Monday-Friday 2:30-6:30 in MC 106.

Midterm average: 53/70 = 76%

Question: If $P$ is invertible, how do $\det A$ and $\det(P^{-1}AP)$ compare?

Partial review of last class: Section 4.3

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.

Finding eigenvalues and eigenspaces: Let $A$ be an $n \times n$ matrix.

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 $(A - \lambda I) \vx = \vec 0$.

So we need to get good at solving polynomial equations. Solutions are called zeros or roots.

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 $g$).

New material: 4.3 continued

Example 4.18: Find the eigenvalues and eigenspaces of $A = \bmat{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4 \emat$.

Solution: 1. Last time, we computed the characteristic polynomial: $$\det (A - \lambda I) = - \lambda^3 + 4 \lambda^2 - 5 \lambda + 2$$ 2. Then we found that $$- \lambda^3 + 4 \lambda^2 - 5 \lambda + 2 = - (\lambda - 1)^2 (\lambda - 2)$$ with roots $\lambda = 1$ and $\lambda = 2$.

3. To find the $\lambda = 1$ eigenspace, we do row reduction: $$\kern-4ex [A - I \mid 0\,] = \bmat{rrr|r} -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 2 & -5 & 3 & 0 \emat \lra{} \bmat{rrr|r} 1 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \emat$$ We find that $x_3 = t$ is free and $x_1 = x_2 = x_3$, so $$E_1 = \left\{ \colll t t t \right\} = \span \left( \colll 1 1 1 \right)$$ So $\colll 1 1 1$ is a basis of the eigenspace corresponding to $\lambda = 1$. Check!

Finding a basis for $E_2$ is similar; see text.

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.

In the previous example, $\lambda = 1$ has algebraic multiplicity 2 and $\lambda = 2$ has algebraic multiplicity 1.

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 geometric multiplicity 1 (and so does $\lambda = 2$).

Example 4.19: Find the eigenvalues and eigenspaces of $A = \bmat{rrr} -1 & 0 & 1 \\ 3 & 0 & -3 \\ 1 & 0 & -1 \emat$. 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 $A$

Next: how to become a Billionaire using the material from this course.