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5. Similarity
I. Complex Vector Spaces
II. Similarity
III. Nilpotence
IV. Jordan Form
Topics
Goal: Given H = hB→B , find D s.t. K = hD→D has a simple (Jordan) form.
5.I. Complex Vector Spaces
A complex vector space is a linear space with complex numbers as scalars,
i.e., the scalar multiplication is over C, the complex number field.
All n-D complex vector spaces are isomorphic to Cn.
Motivation for using complex numbers:
Roots of real algebra equations can be complex.
5.I.1.
Factoring and Complex Numbers; A Review
5.I.2.
Complex Representations
5.I.1. Factoring and Complex Numbers; A Review
Theorem 1.1:
Division Theorem for Polynomials
Let c(x) be a polynomial.
If m(x) is a non-zero polynomial then there are quotient and remainder
polynomials q(x) and r(x) such that
c(x) = m(x) q(x) + r(x)
where the degree of r(x) is strictly less than the degree of m(x).
A constant is a polynomial of degree 0.
Example 1.2:
If c(x) = 2x3  3x2 + 4x and m(x) = x2 +1 then
q(x) = 2x  3
and
r(x) = 2x + 3
Note that r(x) has a lower degree than m(x).
Corollary 1.3:
The remainder when c(x) is divided by x λ, is the constant polynomial r(x) = c(λ).
Proof:
Setting m(x) = x λin c(x) = m(x) q(x) + r(x) gives
c(λ) = (λ λ) q(λ) + r(x) = r(x)
QED
Definition:
Let
c(x) = m(x) q(x) + r(x)
Then m(x) is a factor of c(x) if r(x) = 0.
Corollary 1.4:
If λ is a root of the polynomial c(x) then x λ divides c(x) evenly,
i.e., x λis a factor of c(x).
Quadratic formula:
The roots of a x2 + b x + c are
b  b2  4ac
 
2a
D = b2  4ac is the discriminant.
D < 0 → λ are complex conjugates.
A polynomial that cannot be factored into two lower-degree polynomials
with real number coefficients is irreducible over the reals.
Theorem 1.5:
Any constant or linear polynomial is irreducible over the reals.
A quadratic polynomial is irreducible over the reals iff its discriminant is negative.
No cubic or higher-degree polynomial is irreducible over the reals.
Corollary 1.6:
Any polynomial with real coefficients can be factored into linear and irreducible
quadratic polynomials.
This factorization is unique; any two factorizations have the same powers of
the same factors.
Example 1.7:
Because of uniqueness we know, without multiplying them out, that
(x + 3)2 (x2 + 1)3  (x + 3)4 (x2 + x + 1)2
Example 1.8:
By uniqueness, if c(x) = m(x) q(x) then
where c(x) = (x  3)2(x + 2)3 and m(x) = (x  3)(x + 2)2,
we know that q(x) = (x  3)(x + 2).
A complex number is a number a + b i , where a, b  R and i 2 = 1.
The set C of all complex numbers is a field { C , +,  }, where
(a+bi)+(c+di)(a+c)+(b+d)i
(a+bi)(c+di)(acbd)+(ad+bc)i
Corollary 1.10: Fundamental Theorem of Algebra
Polynomials with complex coefficients factor into linear polynomials with
complex coefficients. This factorization is unique.
5.I.2. Complex Representations
Example 2.2:
2  1 1   1  i  6i 1  i  2i 
1  i

 i


2  3i  3i i   i  6i  9 i  2i  3 

Standard basis of Cn is
En 
1
 0
 ,
 
 
 0
 0
 0
, 
 
 
1
 1  7i 1  i 


 9  5i 3  3i 