Polynomials

Introduction to Complex Numbers

Notation

The set $𝔽$ refers to either the field $ℝ$ or the field $ℂ$ .

Definition 82: Complex Conjugate

Let $𝑧 = 𝑎 + 𝑏 𝑖 \in ℂ$ be a complex number, where $𝑎, 𝑏 \in ℝ$ . The complex conjugate of $𝑧$ , denoted $\bar{𝑧}$ , is defined as:

\bar{𝑧} = 𝑎 - 𝑏 𝑖 .

Property: Properties of Complex Numbers

Double Conjugation. $\bar{\bar{𝑧}} = 𝑧$ .
Distributivity of Conjugation. $\bar{𝑧 + 𝑤} = \bar{𝑧} + \bar{𝑤}$ and $\bar{𝑧 𝑤} = \bar{𝑧} \bar{𝑤}$ .
Conjugate of a Quotient. If $𝑤 \neq 0$ , then $\bar{𝑧 / 𝑤} = \bar{𝑧} / \bar{𝑤}$ .
Conjugate of a Power. $\bar{𝑧^{𝑛}} = {\bar{𝑧}}^{𝑛}$ for all $𝑛 \in ℤ^{+}$ .
Conjugate of a Real Number. If $𝑧 \in ℝ$ , then $\bar{𝑧} = 𝑧$ .
Sum of Conjugates. $𝑧 + \bar{𝑧} = 2 Re (𝑧)$ .
Product of Conjugates. $𝑧 \bar{𝑧} = | 𝑧 |^{2}$ .

Definition 83: Absolute Value of a Complex Number

Let $𝑧 = 𝑎 + 𝑏 𝑖 \in ℂ$ be a complex number, where $𝑎, 𝑏 \in ℝ$ . The absolute value or the modulus of $𝑧$ , denoted $| 𝑧 |$ , is defined as:

| 𝑧 | = \sqrt{𝑎^{2} + 𝑏^{2}} .

Property

For all $𝑧 = 𝑎 + 𝑏 𝑖 \in ℂ$ , ${| 𝑧 |}^{2} = 𝑧 \bar{𝑧} = 𝑎^{2} + 𝑏^{2}$ .

Introduction to Polynomials

Definition 84: Polynomial

A polynomial over a field $𝐹$ with respect to the indeterminate $𝑥$ is an expression of the form:

𝑝 (𝑥) = 𝑎_{𝑛} 𝑥^{𝑛} + 𝑎_{𝑛 - 1} 𝑥^{𝑛 - 1} + \dots + 𝑎_{1} 𝑥 + 𝑎_{0},

where $𝑛 \in ℤ^{+}$ and $𝑎_{0}, 𝑎_{1}, \dots, 𝑎_{𝑛} \in 𝐹$ are called the coefficients of the polynomial. The degree of the polynomial $𝑝 (𝑥)$ is the largest integer $𝑛$ such that $𝑎_{𝑛} \neq 0$ , denoted $\deg (𝑝 (𝑥)) = 𝑛$ .

That is, an $𝑛$ ^th degree polynomial over $𝐹$ with respect to $𝑥$ is a linear combination of ${1, 𝑥, 𝑥^{2}, \dots, 𝑥^{𝑛}}$ with weights in $𝐹$ , constraining the weight of $𝑥^{𝑛}$ to be nonzero.

Notation

The set of all polynomials over a field $𝐹$ with respect to the indeterminate $𝑥$ is denoted $ℙ (𝐹; 𝑥)$ or $𝐹 [𝑥]$ . The set of all polynomials over a field $𝐹$ with respect to the indeterminate $𝑥$ of degree at most $𝑛$ is denoted $ℙ_{𝑛} (𝐹; 𝑥)$ or $𝐹_{𝑛} [𝑥]$ . The indeterminate $𝑥$ is often omitted when the context is clear, so we may often write just $ℙ (𝐹)$ or $ℙ_{𝑛} (𝐹)$ .

Corollary 38

$ℙ_{𝑛} (𝐹)$ is an $(𝑛 + 1)$ -dimensional vector space over $𝐹$ .

Definition 85: Zero of a Polynomial

A scalar $𝛼 \in 𝐹$ is a zero of the polynomial $𝑝 \in ℙ (𝐹)$ iff $𝑝 (𝛼) = 0$ .

Notation

The set of all polynomials of degree exactly $𝑛$ over a field $𝐹$ is denoted $ℙ_{𝑛}^{=} (𝐹)$ .

Theorem 70: Factorization of Polynomials

Suppose $𝑝 \in ℙ_{𝑛}^{=} (𝐹)$ is a polynomial of degree exactly $𝑛$ and $𝛼 \in 𝐹$ . Then $𝛼$ is a zero of $𝑝$ if and only if there exists a polynomial $𝑞 \in ℙ_{𝑛 - 1}^{=} (𝐹)$ such that:

𝑝 (𝑥) = (𝑥 - 𝛼) 𝑞 (𝑥) .

Characteristic Polynomials

Definition 86: Characteristic Polynomial

Let $𝑇 : 𝑉 \to 𝑉$ be a linear map on an $𝑛$ -dimensional vector space $𝑉$ over a field $𝐹$ . The characteristic polynomial $𝜒$ of $𝑇$ is defined as:

𝜒 (𝜆) = \det (𝜆 {id}_{𝑉} - 𝑇) = \det (𝜆 𝐼_{𝑛} - {[𝑇]}_{ℬ ℬ}),

where $𝐼_{𝑛}$ is the $𝑛 \times 𝑛$ identity matrix and $ℬ$ is any basis for $𝑉$ .

Polynomials Applied to Operators

Theorem 71: Cayley-Hamilton Theorem

Let $𝑇 : 𝑉 \to 𝑉$ be a linear operator on a finite-dimensional vector space $𝑉$ . Then the characteristic polynomial $𝜒$ of $𝑇$ annihilates $𝑇$ , i.e.:

𝜒 (𝑇) = 0 .

Polynomials

Introduction to Complex Numbers

Notation

Definition 82: Complex Conjugate

Property: Properties of Complex Numbers

Definition 83: Absolute Value of a Complex Number

Property

Introduction to Polynomials

Definition 84: Polynomial

Notation

Corollary 38

Definition 85: Zero of a Polynomial

Notation

Theorem 70: Factorization of Polynomials

Characteristic Polynomials

Definition 86: Characteristic Polynomial

Polynomials Applied to Operators

Theorem 71: Cayley-Hamilton Theorem

Minimal Polynomials