ENGG2013 Unit 20 Extensions to Complex numbers Mar, 2011.

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, [a 1 a 2 … a n ], is defined as, which is also called the norm of [a 1 a 2 … a n ]. kshumENGG20132

Examples We usually denote the norm of a vector v by || v ||. kshumENGG20133

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 kshumENGG20134

REVIEW OF COMPLEX NUMBERS kshumENGG20135

Quadratic equation When the discriminant of a quadratic equation is negative, there is no real solution. The complex roots are kshumENGG20136

Complex eigenvalues There are some matrices whose eigenvalues are complex numbers. The characteristic polynomial of this matrix is The eigenvalues are kshumENGG20137

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–2i 2 =4+3i. kshumENGG20138

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) = (a 2 +b 2 ) 1/2. – We use the notation | a+bi | to stand for the absolute value a 2 +b 2. Division: (1+2i)/(2 – i) kshumENGG20139

The complex plane kshumENGG Re Im 1+2i 2 – i 3+i

Polar form kshumENGG Re Im a+bi = r (cos + i sin ) = r e i r a b

COMPLEX MATRICES kshumENGG201312

Complex vectors and matrices Complex vector: vector with complex entries – Examples: Complex matrix: matrix with complex entries kshumENGG201313

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 kshumENGG201314

Definition The norm, or length, of a complex vector [z 1 z 2 … z n ] where z 1, z 2, … z n are complex numbers, is defined as Example – The norm of [i 1] is – The norm of [2i 1] is kshumENGG201315

Complex dot product For complex vector, the dot product is replaced by where c 1, d 1, e 1, c 2, d 2, e 2 are complex numbers and c 1 *, d 1 *, and e 1 * are the conjugates of c 1, d 1, and e 1 respectively. kshumENGG201316

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 M H or. kshumENGG201317

Example kshumENGG Hermitian

Example kshumENGG201319

Complex matrix in special form Hermitian: A H =A. Skew-Hermitian: A H = –A. Unitary: A H =A -1, or equivalently A H A = I. Example: kshumENGG201320

Charles Hermite Dec 24, 1822 – Jan 14, French mathematician Introduced the notion of Hermitian operator Proved that the base of the natural log, e, is transcendental. kshumENGG

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 v 1, v 2, …, v n, are called orthonormal if they are (i) mutually orthogonal v i H v j =0 for i j, and (ii) v i H v i =1 for all i. We can find a unitary matrix U, such that M can be written as UDU H, for some diagonal matrix with real diagonal entries. kshumENGG

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 UDU H, for some diagonal matrix with purely imaginary diagonal entries. kshumENGG

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 VDV H, for some diagonal matrix whose diagonal entries lie on the unit circle in the complex plane. kshumENGG

Eigenvalues of Hermitian, skew- Hermitian and unitary matrices kshumENGG Hermitian Re Im Complex plane 1 Skew-Hermitian unitary

Generalization: Normal matrix A complex matrix N is called normal, if N H N = N N H. 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 UDU H, for some diagonal matrix whose diagonal entries are the eigenvalues of N. kshumENGG

COMPLEX EXPONENTIAL FUNCTION kshumENGG201327

Exponential function Definition for real x: kshumENGG y = e x.

Derivative of exp(x) kshumENGG y= e x y=1+x For example, the slope of the tangent line at x=0 is equal to e 0 =1.

Taylor series expansion We extend the definition of exponential function to complex number via this Taylor series expansion. For complex number z, e z is defined by simply replacing the real number x by complex number z: kshumENGG201330

Series expansion of sin and cos Likewise, we extend the definition of sin and cos to complex number, by simply replacing real number x by complex number z. kshumENGG

Example For real number : kshumENGG201332

Eulers formula kshumENGG For real number, Proof:

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. kshumENGG201334