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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, [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. 10 9 • The complex roots are 8 7 y 6 5 4 3 2 1 0 -2 kshum ENGG2013 -1 0 1 x 2 3 4 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 kshum ENGG2013 18 Example kshum ENGG2013 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 • 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 http://en.wikipedia.org/wiki/Charles_Hermite Charles Hermite 21 Properties of Hermitian matrix Let M be an nn 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 nn 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 nn 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, skewHermitian and unitary matrices Im Complex plane Hermitian Skew-Hermitian kshum ENGG2013 Re 1 25 Generalization: Normal matrix A complex matrix N is called normal, if NH N = N N H . • Normal matrices contain symmetric, skewsymmetric, orthogonal, Hermitian, skewHermitain 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. 8 7 6 y 5 4 3 2 1 0 -2 -1.5 -1 -0.5 0 x 0.5 1 1.5 2 http://en.wikipedia.org/wiki/Exponential_function kshum ENGG2013 28 Derivative of exp(x) y= ex 3 2 1 y For example, the slope of the tangent line at x=0 is equal to e0=1. 0 -1 -2 -3 -3 kshum -2 ENGG2013 -1 0 x 1 2 3 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 http://en.wikipedia.org/wiki/Taylor_series 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. kshum ENGG2013 31 Example • For real number : kshum ENGG2013 32 Euler’s formula For real number , Proof: kshum ENGG2013 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