5. Similarity I.Complex Vector Spaces II.Similarity III.Nilpotence IV.Jordan Form Topics Goal: Given H = h B → B, find D s.t. K = h D → D has a simple (Jordan) form.

5.I. Complex Vector Spaces Motivation for using complex numbers: Roots of real algebra equations can be complex. 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 C n. 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) = 2x 3  3x 2 + 4x and m(x) = x 2 +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 x 2 + b x + c are D = b 2  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 (x 2 + 1) 3  (x + 3) 4 (x 2 + 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).

Corollary 1.10:Fundamental Theorem of Algebra Polynomials with complex coefficients factor into linear polynomials with complex coefficients. This factorization is unique. 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 + b i ) + ( c + d i )  ( a + c ) + ( b + d ) i ( a + b i )  ( c + d i )  ( a c  b d ) + ( a d + b c ) i

5.I.2.Complex Representations Example 2.2: Standard basis of C n is