Extensions to Complex numbers 1

Definition: Norm of a vector By Pythagoras theorem, the length of a vector with two components [a b] is The length of a vector with three components [a b c] is The length of a vector with n components, [a1 a2 … an], is defined as , which is also called the norm of [a1 a2 … an]. kshum ENGG2013 2

Examples We usually denote the norm of a vector v by || v ||. kshum ENGG2013 3

Norm squared The square of the norm, or square of the length, of a column vector v can be conveniently written as the dot product Example kshum ENGG2013 4

REVIEW OF COMPLEX NUMBERS kshum ENGG2013 5

Quadratic equation When the discriminant of a quadratic equation is negative, there is no real solution. The complex roots are kshum ENGG2013 6

Complex eigenvalues There are some matrices whose eigenvalues are complex numbers. The characteristic polynomial of this matrix is The eigenvalues are kshum ENGG2013 7

Complex numbers Let i be the square root of –1. A complex number is written in the form a+bi where a and b are real numbers. “a” is called the “real part” and “b” is called the “imaginary part” of a+bi. Addition: (1+2i) + (2 – i) = 3+i. Subtraction: (1+2i) – (2 – i) = –1 + 3i. Multiplication: (1+2i)(2 – i) = 2+4i–i–2i2=4+3i. kshum ENGG2013 8

Complex numbers The conjugate of a+bi is defined as a – bi. The absolute value of a+bi is defined as (a+bi)(a – bi) = (a2+b2)1/2. We use the notation | a+bi | to stand for the absolute value a2+b2. Division: (1+2i)/(2 – i) kshum ENGG2013 9

The complex plane Im 1+2i 3+i Re 2 – i kshum ENGG2013 10

Polar form Im a+bi = r (cos  + i sin ) = r ei a r  Re b kshum ENGG2013 11

COMPLEX MATRICES kshum ENGG2013 12

Complex vectors and matrices Complex vector: vector with complex entries Examples: Complex matrix: matrix with complex entries kshum ENGG2013 13

Length of complex vector If we apply the calculation of the length of a vector to a complex, something strange may happen. Example: the “length” of [i 1] would be Example: the “length” of [2i 1] would be kshum ENGG2013 14

Definition The norm, or length, of a complex vector [z1 z2 … zn] where z1, z2, … zn are complex numbers, is defined as Example The norm of [i 1] is The norm of [2i 1] is kshum ENGG2013 15

Complex dot product For complex vector, the dot product is replaced by where c1, d1, e1, c2, d2, e2 are complex numbers and c1*, d1*, and e1* are the conjugates of c1, d1, and e1 respectively. kshum ENGG2013 16

The Hermitian operator The transpose operator for real matrix should be replaced by the Hermitian operator. The conjugate of a vector v is obtained by taking the conjugate of each component in v. The conjugate of a matrix M is obtained by taking the conjugate of each entry in M. The Hermitian of a complex matrix M, is defined as the conjugate transpose of M. The Hermitian of M is denoted by MH or . kshum ENGG2013 17

Example Hermitian \|\mathbf{v}\|^2 = \mathbf{v}^H\mathbf{v} = \begin{bmatrix} 1& -4i& 2-i \end{bmatrix}\begin{bmatrix} 1\\ 4i\\ 2+i \end{bmatrix} = 1+16+5 = 22 kshum ENGG2013 18 18

Example \begin{bmatrix} 2 & 2i & 1-i \\ -2i &0 & i\\ 1+i & -i & 1 \end{bmatrix}^H =\begin{bmatrix} \end{bmatrix} kshum ENGG2013 19 19

Complex matrix in special form Hermitian: AH=A. Skew-Hermitian: AH= –A. Unitary: AH =A-1, or equivalently AH A = I. Example: kshum ENGG2013 20

Charles Hermite Dec 24, 1822 – Jan 14, 1901. French mathematician http://en.wikipedia.org/wiki/Charles_Hermite Dec 24, 1822 – Jan 14, 1901. French mathematician Introduced the notion of Hermitian operator Proved that the base of the natural log, e, is transcendental. kshum ENGG2013 21

Properties of Hermitian matrix Let M be an nn complex Hermitian matrix. The eigenvalues of M are real numbers. We can choose n orthonormal eigenvectors of M. n vectors v1, v2, …, vn, are called “orthonormal” if they are (i) mutually orthogonal viH vj =0 for i j, and (ii) viH vi =1 for all i. We can find a unitary matrix U, such that M can be written as UDUH, for some diagonal matrix with real diagonal entries. http://en.wikipedia.org/wiki/Hermitian_matrix kshum ENGG2013 22

Properties of skew-Hermitian matrix Let S be an nn complex skew-Hermitian matrix. The eigenvalues of S are purely imaginary. We can choose n orthonormal eigenvectors of S. We can find a unitary matrix U, such that S can be written as UDUH, for some diagonal matrix with purely imaginary diagonal entries. http://en.wikipedia.org/wiki/Skew-Hermitian_matrix kshum ENGG2013 23

Properties of unitary matrix Let U be an nn complex unitary matrix. The eigenvalues of U have absolute value 1. We can choose n orthonormal eigenvectors of U. We can find a unitary matrix V, such that U can be written as VDVH, for some diagonal matrix whose diagonal entries lie on the unit circle in the complex plane. http://en.wikipedia.org/wiki/Unitary_matrix kshum ENGG2013 24

Eigenvalues of Hermitian, skew-Hermitian and unitary matrices Im Complex plane unitary Hermitian Re Skew-Hermitian 1 kshum ENGG2013 25

Generalization: Normal matrix A complex matrix N is called normal, if NH N = N NH. Normal matrices contain symmetric, skew- symmetric, orthogonal, Hermitian, skew- Hermitain and unitary as special cases. We can find a unitary matrix U, such that N can be written as UDUH, for some diagonal matrix whose diagonal entries are the eigenvalues of N. http://en.wikipedia.org/wiki/Normal_matrix kshum ENGG2013 26

COMPLEX EXPONENTIAL FUNCTION kshum ENGG2013 27

Exponential function Definition for real x: y = ex. http://en.wikipedia.org/wiki/Exponential_function kshum ENGG2013 28

Derivative of exp(x) y= ex y=1+x For example, the slope of the tangent line at x=0 is equal to e0=1. e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1+x+\frac{x^2}{2}+\frac{x^3}{3!}+ \ldots. kshum ENGG2013 29 29

Taylor series expansion We extend the definition of exponential function to complex number via this Taylor series expansion. For complex number z, ez is defined by simply replacing the real number x by complex number z: kshum ENGG2013 30

Series expansion of sin and cos http://en.wikipedia.org/wiki/Taylor_series Likewise, we extend the definition of sin and cos to complex number, by simply replacing real number x by complex number z. kshum ENGG2013 31

Example For real number : \begin{align*} cos(i\theta) &= 1-\frac{(i\theta)^2}{2!}+\frac{(i\theta)^4}{4!}- \frac{(i\theta)^6}{6!}+\ldots\\ &= 1+ \frac{\theta^2}{2!}+\frac{\theta^4}{4!}+\frac{\theta^6}{6!}+\ldots \end{align*} kshum ENGG2013 32 32

Euler’s formula For real number , Proof: kshum ENGG2013 33 33

Summary Matrix and vector are extended from real to complex Transpose  conjugate transpose (Hermitian operator) Symmetric  Hermitian Skew-symmetric  skew-Hermitian Exponential function and sinusoidal function are extended from real to complex by power series. kshum ENGG2013 34