1 Matrix Algebra and Random Vectors Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia

2 General Statistical Distance

3

4 Quadratic Form

5 Statistical Distance under Rotated Coordinate System

6 Rotated Coordinate System 

7 Coordinate Transformation

8 Quadratic Form in Transformed Coordinate Systems

9 Diagonalized Quadratic Form

10 Orthogonal Matrix

11 Diagonalization

12 Concept of Mapping x

13 Eigenvalues

14 Eigenvectors

15 Spectral Decomposition

16 Positive Definite Matrix Matrix A is non-negative definite if for all x ’ =[ x 1, x 2, …, x k ] for all x ’ =[ x 1, x 2, …, x k ] Matrix A is positive definite if for all non-zero x for all non-zero x

17 Positive Definite Matrix

18 Inverse and Square-Root Matrix

19 Random Vectors and Random Matrices Random vector –Vector whose elements are random variables Random matrix –Matrix whose elements are random variables

20 Expected Value of a Random Matrix

21 Population Mean Vectors

22 Covariance

23 Statistically Independent

24 Population Variance-Covariance Matrices

25 Population Correlation Coefficients

26 Standard Deviation Matrix

27 Correlation Matrix from Covariance Matrix

28 Partitioning Covariance Matrix

29 Partitioning Covariance Matrix

30 Linear Combinations of Random Variables

31 Example of Linear Combinations of Random Variables

32 Linear Combinations of Random Variables

33 Sample Mean Vector and Covariance Matrix

34 Partitioning Sample Mean Vector

35 Partitioning Sample Covariance Matrix

36 Cauchy-Schwarz Inequality

37 Extended Cauchy-Schwarz Inequality

38 Maximization Lemma

39 Maximization of Quadratic Forms for Points on the Unit Sphere

40 Maximization of Quadratic Forms for Points on the Unit Sphere

41 Maximization of Quadratic Forms for Points on the Unit Sphere

42 Maximization of Quadratic Forms for Points on the Unit Sphere