Today we finish 4.2 and start 4.3. Continue **reading** Section
4.3 for Friday
and also read **Appendix D** on polynomials (**self-study**).

Work through recommended homework questions.

**Tutorials:** No quiz this week, just review.

**Office hour:** Wednesday, 12:30-1:30, MC103B.

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

**Midterm 2 Solutions** are available from the course home page. I don't know the
class average yet.

**Question:** True/false: If $A$ is not invertible, then $AB$ is not
invertible.

**Question:** True/false: $\det(A+B) = \det A + \det B$.

**Question:** $\det (3 I_2) = \query{3^2 \det I_2 = 3^2 = 9}$

**Question:** $\bdmat{rrr} 0 & 0 & a \\ 0 & b & c \\ d & e & f \edmat
= \query{-\bdmat{rrr} d & e & f \\ 0 & b & c \\ 0 & 0 & a \edmat = -abd \qtext{(not triangular!)}}$

**Notation:** If $A$ is an $n \times n$ matrix and $\vb \in \R^n$, we write
$A_i(\vb)$ for the matrix obtained from $A$ by replacing the $i$th column with
the vector $\vb$:
$$
A_i(\vb) = [ \va_1 \cdots \va_{i-1} \vb \, \va_{i+1} \cdots \va_n \,]
$$

**Theorem:** Let $A$ be an invertible $n \times n$
matrix and let $\vb$ be in $\R^n$. Then the unique solution $\vx$ of the
system $A \vx = \vb$ has components
$$
x_i = \frac{\det(A_i(\vb))}{\det A},\quad \text{for } i = 1, \ldots, n
$$

**Theorem:** If $A$ is an invertible matrix, then
$$
A^{-1} = \frac{1}{\det A} \adj A
$$

**Example:** If $A = \bmat{rr} a & b \\ c & d \emat$, then
the cofactors are
$$
\begin{aligned}
C_{11} &= + \det [d] = +d & C_{12} &= - \det [c] = -c \\
C_{21} &= - \det [b] = -b & C_{22} &= + \det [a] = +a \\
\end{aligned}
$$
so the adjoint matrix is
$$
\adj A = \bmat{rr} d & -b \\ -c & a \emat
$$
and
$$
A^{-1} = \frac{1}{\det A} \adj A = \frac{1}{\det A} \bmat{rr} d & -b \\ -c & a \emat
$$
as we saw before.

See Example 4.17 in the text for a $3 \times 3$ example. Again, this is not generally a good computational approach. It's importance is theoretical.

**Definition:** Let $A$ be an $n \times n$ matrix.
A scalar $\lambda$ (lambda) is called an **eigenvalue** of $A$ if
there is a nonzero vector $\vx$ such that $A \vx = \lambda \vx$.
Such a vector $\vx$ is called an **eigenvector** of $A$ corresponding to $\lambda$.

The eigenvectors for a given eigenvalue $\lambda$ are
the **nonzero** solutions to $(A - \lambda I) \vx = \vec 0$.

**Definition:** The collection of **all** solutions to
$(A - \lambda I) \vx = \vec 0$ is a subspace called the
**eigenspace** of $\lambda$ and is denoted $E_\lambda$. In other words,
$$ E_\lambda = \null(A - \lambda I) . $$
It consists of the eigenvectors plus the zero vector.

By the fundamental theorem of invertible matrices, $A - \lambda I$ has a nontrivial null space if and only if it is not invertible, and we now know that this is the case if and only if $\det (A - \lambda I) = 0$.

The expression $\det (A - \lambda I)$ is always a polynomial in $\lambda$. For example, when $A = \bmat{rr} a & b \\ c & d \emat$, $$ \kern-8ex \det(A- \lambda I) = \bdmat{cc} a-\lambda & b \\ c & d-\lambda \edmat = (a - \lambda)(d-\lambda) - bc = \lambda^2 - (a+d)\lambda + (ad - bc) $$ In $A$ is $3 \times 3$, then $$ \kern-8ex \det(A - \lambda I) = (a_{11} - \lambda) \bdmat{cc} a_{22} - \lambda\! & a_{23} \\ a_{32} & \!a_{33} - \lambda \edmat - a_{12} \bdmat{cc} a_{21} & a_{23} \\ a_{31} & \!a_{33} - \lambda \edmat + a_{13} \bdmat{cc} a_{21} & \!a_{22} -\lambda \\ a_{31} & a_{32} \edmat $$ which is a degree 3 polynomial in $\lambda$.

Similarly, 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**.

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 $E_\lambda = \null (A - \lambda I)$
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. On whiteboard, compute the characteristic polynomial:
$$
\det (A - \lambda I) = - \lambda^3 + 4 \lambda^2 - 5 \lambda + 2
$$
2. To find the roots, it is often worth trying a few small integers
to start. We see that $\lambda = 1$ works. So by the factor theorem,
we know $\lambda - 1$ is a factor:
$$
- \lambda^3 + 4 \lambda^2 - 5 \lambda + 2
= (\lambda - 1)(\query{-} \lambda^2 + \query{3} \lambda \toggle{+ \text{?}}{-2}\endtoggle)
$$
Now we need to find roots of $-\lambda^2 + 3 \lambda - 2$.
Again, $\lambda = 1$ works, and this factors as $-(\lambda - 1)(\lambda - 2)$.
So
$$
\det (A - \lambda I) = - \lambda^3 + 4 \lambda^2 - 5 \lambda + 2
= - (\lambda - 1)^2 (\lambda - 2)
$$
and the roots are $\lambda = 1$ and $\lambda = 2$.

3. To find the $\lambda = 1$ eigenspace, we do row reduction: $$ [A - I \mid 0\,] = \bmat{rrr|r} -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 2 & -5 & 3 & 0 \emat \lra{} \bmat{rrr|r} -1 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \emat $$ 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 $\lambda = 1$.

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 whiteboard.

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.