Lecture Notes for 120 - UCLA Department of Mathematics

A.2. GEOMETRY 174
the characteristic polynomial is
so the discriminant is
2
(a + d) + ad b 2
=(a + d) 2 4 ad b 2 =(a d) 2 +4b 2 0
This shows that the roots must be real.
For a 3 ⇥ 3 matrix the characteristic polynomial is cubic. The intermediate
value theorem then guarantees at least one real root. If we make a change of basis
to another orthonormal basis where the first basis vector is an eigenvector then the
new matrix will still be symmetric and look like
2
4 1 0 0
0 a b
0 b d
The characteristic polynomial then looks like
( 1)
2
3
5
(a + d) + ad b 2
where we see as before that
2
(a + d) + ad b 2 has two real roots. ⇤
A.2. Geometry
Here are a few geometric formulas that use vector notation:
• The length or size of a vector X is denoted:
|X| = p X t · X
• The distance from X to a point P :
|X P |
• The projection of a vector X onto another vector N:
X · N
|N| 2 N
• The signed distance from P to a plane that goes through X 0 and has
normal N, i.e.,givenby(X X 0 ) · N =0:
(P X 0 ) · N
|N|
the actual distance is the absolute value of the signed distance. This
formula also works for the (signed) distance from a point to a line in R 2 .
• The distance from P to a line with direction N that passes through X 0 :
s
(P X 0 ) · N
(P X 0 )
|N| 2 N = |P X 0 | 2 |(P X 0 ) · N| 2
|N| 2
• The area of a parallelogram spanned by two vectors X, Y is
r ⇣ ⇥ ⇤ t ⇥ ⇤ ⌘
det X Y X Y
• If X, Y 2 R 2 there is also a signed area given by
det ⇥ X Y ⇤