1. 8.1 Vector spaces A set of vector is said to form a linear vector space V Chapter 8 Matrices and vector spaces

2. Basis vector The inner product is defined by

3. Chapter 8 Matrices and vector spaces some properties of inner product

4. Chapter 8 Matrices and vector spaces

5. Some useful inequalities: Chapter 8 Matrices and vector spaces

8. 8.2 Linear operators The action of operator A is independent of any basis or coordinate system.

9. Chapter 8 Matrices and vector spaces Properties of linear operators:

10. Chapter 8 Matrices and vector spaces 8.3 Matrices

11. Chapter 8 Matrices and vector spaces 8.4 Basic matrix algebra Matrix addition Multiplication by a scalar

12. Chapter 8 Matrices and vector spaces Multiplication of matrices

13. Chapter 8 Matrices and vector spaces

14. Functions of matrices The transpose of a matrix

15. Chapter 8 Matrices and vector spaces For a complex matrix

16. Chapter 8 Matrices and vector spaces If the basis is not orthonormal The trace of a matrix

17. Chapter 8 Matrices and vector spaces 8.9 The determinant of a matrix

18. Chapter 8 Matrices and vector spaces Ex: Matrix A is 3×3, for three 3-D vectors Properties of determinants

19. Chapter 8 Matrices and vector spaces

20. Ex: Evaluate the determinant (2)+(3) put (3) (4)-(2) put (4)

21. The elements of the inverse matrix are Chapter 8 Matrices and vector spaces 8.10 The inverse of a matrix

22. Chapter 8 Matrices and vector spaces Useful properties:

23. Chapter 8 Matrices and vector spaces 8.12 Special types of square matrix Diagonal matrices

24. Chapter 8 Matrices and vector spaces Lower triangular matrix Upper triangular matrix Symmetric and antisymmetric matrices

25. Chapter 8 Matrices and vector spaces Ex: If A is N×N antisymmetric matrix, show that |A|=0 if N is odd. Orthogonal matrices Hermitian and anti-Hermitian matrices

26. Chapter 8 Matrices and vector spaces Unitary matrices Normal matrices

27. Chapter 8 Matrices and vector spaces 8.13 Eigenvectors and eigenvalues Ex: A non-singular matrix A has eigenvalues λ i, and eigenvectors x i. Find the eigenvalues and eigenvectors of the inverse matrix A -1.

28. Chapter 8 Matrices and vector spaces Eigenvectors and eigenvalues of a normal matrix

29. Chapter 8 Matrices and vector spaces An eigenvalue corresponding to two or more different eigenvectors is said to be degenerate.

30. Chapter 8 Matrices and vector spaces

31. Ex: Show that a normal matrix can be written in terms of its eigenvalues and orthogonal eigenvectors as Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices

32. Chapter 8 Matrices and vector spaces Ex: Prove that the eigenvectors corresponding to different eigenvalues of an Hermitian matrix are orthogonal. anti-Hermitian matrix

33. Chapter 8 Matrices and vector spaces Unitary matrix Simultaneous eigenvectors

34. Chapter 8 Matrices and vector spaces

35. 8.14 Determination of eigenvalues and eigenvectors Ex: Find the eigenvalues and normalized eigenvectors of the real symmetric matrix

36. Chapter 8 Matrices and vector spaces Sol:

37. Chapter 8 Matrices and vector spaces

38. Degenerate eigenvalues

39. Chapter 8 Matrices and vector spaces 8.15 Change of basis and similarity transformations

40. Chapter 8 Matrices and vector spaces Similarity transformations similarity transformation Properties of the linear operator under two basis

41. Chapter 8 Matrices and vector spaces If S is a unitary matrix:

42. Chapter 8 Matrices and vector spaces 8.16 Diagonalization of matrices

43. Chapter 8 Matrices and vector spaces For normal matrices (Hermitian, anti-Hermitian and unitary) the N eigenvectors are linear independent.

44. Chapter 8 Matrices and vector spaces Ex: Prove the trace formula

45. Chapter 8 Matrices and vector spaces 8.17 Quadratic and Hermitian forms

46. Chapter 8 Matrices and vector spaces

49. The stationary properties of the eigenvectors. What are the vector that form a maximum or minimum of

50. Chapter 8 Matrices and vector spaces Ex: Show that if is stationary then is an eigenvector of and is equal to the corresponding eigenvalues.

51. Chapter 8 Matrices and vector spaces 8.18 Simultaneous linear equation

52. Chapter 8 Matrices and vector spaces

56. Ex: Use Cramer’s rule to solve the set of the simultaneous equation.