ENGG2012B Lecture 12 Complex vectors and complex matrices

1. ENGG2012BLecture 12Complex vectors andcomplex matrices Kenneth Shum ENGG2012B

2. Last lecture • Given a square matrix A, a non-zero vector v is called an eigenvector of A, if we can find a real number , such that . • The number  is the eigenvalue of A corresponding to the eigenvector v. Last time, the eigenvalues are all real numbers, because we haven’t introduced complex vectors and matrices yet. ENGG2012B

3. Steps in calculating eigenvalues and eigenvectors • Given a matrix M. • Find the characteristic polynomial. • Find the roots of the characteristic polynomial. • For each eigenvalue  of M, find the non-zero vectors v such that Mv = v. ENGG2012B

4. Example: flip • A linear transformation L(x,y) given by: L(x,y) = (x, -y) x x y – y ENGG2012B

5. Example: shear action • A linear transformation given by L(x,y) = (x+0.25y, y) x x+ 0.25 y y y ENGG2012B

6. Repeated eigenvalues, one linearly independent eigenvector • What are the eigenvalues of ? • Eigenvectors ? • Solve ( k nonzero ) ENGG2012B

7. Example: Expansion • L(x,y) = (ax, ay), for some constant a. x  ax y  ay ENGG2012B

8. Repeated eigenvalues, two linearly independent eigenvectors • What are the eigenvalues of ? • Eigenvectors ? • Solve All non-zerovectors areeigenvector. ENGG2012B

9. Example: Rotation • Rotation by 90 degrees counter-clockwise: L(x,y) = (– y , x). x – y y  x ENGG2012B

10. Eigenvalues = ? No real root ENGG2012B

11. Extension to complex vectors and matrix • Given a square matrix A, a non-zero vector v is called an eigenvector of A, if we an find a number, which may be complex, such that • This number  is the eigenvalue of A corresponding to the eigenvector v. ENGG2012B

12. Complex Eigenvalues ENGG2012B

13. REVIEW OF COMPLEX NUMBERS ENGG2012B

14. The ``worlds’’ of real numbers and complex numbers 1+2j 1/7 1/7 3 -2   -3+j/2 The world of real numbers We can add, subtract, multiplyand divide real numbers,the results are also real numbers The world of complex numbersWe throw in one symbol j, which satisfies j 2 = – 1, and mix with the real numbers ENGG2012B

15. Geometry (Cartesian form) 3 2+3j 2 Imaginary part 1 -1 1 2 3 Real part -1 -2 -3 3-4j -4 ENGG2012B

16. Geometry (Polar form) 3 3.605656.31 3.6056 2 1 56.31 -1 1 2 3 -53.13 -1 -2 5 -3 5 -53.13 -4 ENGG2012B

17. Two equivalent forms of complex numbers • Cartesian form: z = a+ b j • a is the called the real part. • b is the called the imaginary part. • Notations: a = Re(z), b = Im(z) • Polar form: z = r • ris called the magnitude. •  is called the argument. • Notations: r = |z|,  = arg(z) ENGG2012B

18. Arithmetic • Addition: (2+3j) + (3-4j) = 5 – j • Subtraction: (2+3j) – (3 – 4j) =– 1 +7j • Multiplication in Cartesian form: (2+3j)(3 – 4j) = 23 + 3j(–4j) +2(–4)j+3(3j)= 6 –12j2 –8j +9j = 18 + j • Division in Cartesian form: ENGG2012B

19. Addition of complex numbers Im 1+2i 3+i Re 2 – i ENGG2012B

20. Arithmetic (cont’d) • Multiplication in polar form (3.605656.31) (5 – 53.13) = 3.6056 5 (56.31+(–53.13)) =18.028  3.18  // This equals 18+j • Division in polar form (3.605656.31) /(5 – 53.13) = (3.6056/ 5) (56.31–(– 53.13)) = 0.721  109.44  // This equals (-6+17j)/25 ENGG2012B

21. Geometry of multiplication of complex numbers Im r1 r2 When we multiply two complexnumbers, we multiply the magnitudesand add the arguments. r2 1 +2 r1 2 1 Re ENGG2012B

22. A motivation for using complex numbers • The fundamental theorem of algebra: “In the world of complex numbers, a polynomial of degree d has precisely d roots, with multiplicity counted appropriately.” ENGG2012B

23. Conjugate of a complex number 3 z =2+3j 2 Imaginary part 1 -1 1 2 3 Real part -1 We will write z* as the conjugate of a complex number z -2 z* = 2-3j -3 -4 ENGG2012B

24. Conjugate of a complex number • Let p(x) be a polynomial with real coefficients. If z is a complex root of p(x), then z* is also a root of p(x). • Example: Try to solve x2+x+1=0. If only real solutions are allowed, then there is no solution. If complex solutions are allowed, then the solutions are : • The magnitude of a complex number can be expressed in terms of the conjugate as |z|= (z*z)1/2 • Example |2+3j|2 = (2+3j)(2-3j) = 4+9 = 13. ENGG2012B

25. COMPLEX VECTORS AND MATRICES ENGG2012B

26. Complex vectors and matrices • Complex vectors are vectors whose components are complex numbers. • Complex matrices are matrices whose components are complex numbers. • Example of a complex matrix: • Most calculations of vectors and matrices, including Gaussian elimination, determinant, rref, matrix inverse carry over to the complex case. ENGG2012B

27. Euclidean norm of real vector • The magnitude, or norm, of a real vector is defined as the square root of the sum of the squares of the components. • The magnitude of vector (5,12,13) is (52+122+132)1/2= (338)1/2 = 18.385. • In terms of vector transpose, we can write the norm of a column vector v as ENGG2012B

28. Euclidean norm of complex vector • The magnitude of vector (5,12,13j) is (52+122+(13j)2)1/2= (25+144-169)1/2 = 0 ?? • Let Then v is not the zero vector, but vTv is zero. ENGG2012B

29. Euclidean norm of complex vector • The correct way to define the magnitude or norm of a complex vector is the square root of the sum of the squares of the absolute value of the components. • The magnitude of vector (5,12,13j) is (|5|2+|12|2+|13j|2)1/2 =(5 + 144 + 169)1/2 =(338)1/2 • A vector has zero norm only if it is the zero vector. • Define the conjugate of a vector as the vector obtained by taking the conjugate of each component. • Example (1+j, 3-2j)* = (1-j, 3+2j). • In terms of the conjugate of a vector, the norm squared of a column vector v is ENGG2012B

30. Conjugate transpose • Because the operation of conjugate transpose occurs very often in the calculation of complex matrices, the conjugate transpose of a matrix M is called the Hermitian of a matrix • We will use the notation MH for the Hermitian of a matrix. • Example: • For any complex matrices A and B such that AB is well defined, we have (AB)H = BH AH ENGG2012B

31. Symmetric and Hermitian matrix • Given a real matrix A, we say that it is a symmetric matrix if AT = A. • Example of symmetric matrix: • For a complex matrix B, we extend the notion of symmetric matrix, and say that B is Hermitian if BH=B. • Example of Hermitian matrix ENGG2012B

32. Non-Example of Hermitian matrix ENGG2012B

33. Skew-Symmetric and skew-Hermitian matrix • A real matrix A is said to be skew-symmetric if AT = -A. • Example of skew-symmetric matrix: • For a complex matrix B, we say that B is skew-Hermitian if BH = -B. • Example of skew-Hermitian matrix ENGG2012B

34. Charles Hermite (1822-1901) • French mathematician. • He first proved that the constant e, the base of natural log, is transcendental.(A number is transcendental means it is not a root of any polynomial with integer coefficients) http://en.wikipedia.org/wiki/Charles_Hermite ENGG2012B

35. The dot product for complex vector • The dot product, a.k.a. inner product,of two complex vectors (u,v,w) and (x,y,z) is defined as (u,v,w)  (x,y,z)= (u*)x+(v*)y+(w*)z • The dot product of this vector with itself is the magnitude squared of a vector. (u,v,w) (u,v,w)= (u*)u+(v*)v+(w*)w = |u|2 + |v|2 + |w|2. ENGG2012B

36. Orthogonal vectors • Two complex vectors are defined to be perpendicular to each other, or orthogonal, if their dot product is equal to 0. • Example: (1,-2, 4j) and (2, -1, -j) are orthogonal, because1(2) + (-2)(-1) + (-4j)(-j) = 2+2-4 = 0. • Example: (1,-2, 4j) and (2, -1, j) are not orthogonal, because1(2) + (-2)(-1) + (-4j)(j) = 2+2+4 = 8  0. ENGG2012B

37. Orthogonal and unitary matrix • A real matrix A is said to be orthogonal if ATA = I. • Example of orthogonal matrix: • For a complex matrix B, we say that B is unitary if BH B = I. • Example of unitary matrix ENGG2012B

38. Unitary matrix • A complex matrix is unitary if and only if all columns have unit norm, and each pair of columns are orthogonal. • In other words, the columns are orthonormal. Each column has norm 1 M MH I Dot product = 1 ENGG2012B

39. Unitary matrix • A complex matrix is unitary if and only if all columns have unit norm, and each pair of columns are orthogonal. • In other words, the columns are orthonormal. Each pair of distinct columns are orthogonal MH M I Dot product = 0 ENGG2012B

40. Eigenvalues • All eigenvalues of a Hermitian matrix are real numbers • All eigenvalues of a skew-Hermitian matrix are purely imaginary numbers, i.e., their real part is zero. • All eigenvalues of a unitary matrix have absolute value equal to 1, i.e., they lie on the unit circle in the complex plane. ENGG2012B

41. Proof that Hermitian matrix has real eigenvalues • Let M be a Hermitian matrix, i.e., MH=M • Let  be an eigenvalue of M and v be a corresponding eigenvector, i.e,. Mv =  v. • Want to show that  is real, i.e., * = . • Compute vHMv, which is a scalar, in two ways. • vH(Mv) = vH( v) =  vH v • (vH M)v = (MHv)H v = (Mv)H v =(v)H v =* vH v • Because v is a nonzero vector, vH v is positive. From  vH v = * vH v, we get * = . ENGG2012B

42. Eigenvalues of Hermitian, skew-Hermitian and unitary matrices Im Complex plane j unitary Hermitian -1 Re 1 Skew-Hermitian -j ENGG2012B

43. Summary on the three special classes of complex matrices ENGG2012B