Chapter 61 Chapter 7 Review of Matrix Methods Including: Eigen Vectors, Eigen Values, Principle Components, Singular Value Decomposition
Chapter 62 Let x be a dimension vector (using row, column designation for size). It can be expressed as (1)
Chapter 63 Multiplication by a scaler (c) yields (2)
Chapter 64 Addition of two vectors x and y yeilds an vector according to (3)
Chapter 65 Linear Independence (4) A set of vectors are said to be linearly dependent if for k vectors
Chapter 66 In more familiar form, if we can express a vector x i in the following form or (5a) (5b) then x i is linearly dependent on x 1 (5a) or x 1 and x 2 (5b).
Chapter 67 Basis Vectors (6)
Chapter 68 Vector Transpose Then the transpose of M is (7) (8)
Chapter 69 Orthogonality (9)
Chapter 610 Gram Schmidt Process (10) (11) (12)
Chapter 611 scale the Gram Schmidt vectors where (13)
Chapter 612 Matrix Multiplication (14)
Chapter 613 Determinants of Square Matrices In general for a k x k matrix (15) (16)
Chapter 614 nonsingular (17) Is true only if x=0
Chapter 615 Matrix Inverse Note that: (19) (20)
Chapter 616 The inverse of a 2 x 2 matrix A is (20) note must exist for the inverse to exist.
Chapter 617 inverse (21) In general, if B is the inverse of A, then the row (i), column (j) entry in B is given by:
Chapter 618 Trace of a Matrix (22)
Chapter 619 Orthogonality so (23)
Chapter 620 Eigen Vectors and Eigen Values Or equivalently Re-arranging (24) (25) (26)
Chapter 621 i.e. (27) This implies that (28)
Chapter 622 Equation 28 expanded into a polynomial yielding (29) (30)
Chapter 623 Example then or yielding (31) (32) (33)
Chapter 624 Solve for eigen vectors Yielding for Equation 34 (34) (35) (36) (37)
Chapter 625 (38) e 11 must equal 0 and e 12 can take on any value so, we choose 1 for convenience, yielding For Equation 35, we obtain (39) (40)
Chapter 626 e 21 can take on any value choose 1 for convenience, then or yielding (41) (42) (43)
Chapter 627 normalize (44)
Chapter 628 Properties of Symmetric Matrices Note that the covariance matrix(s) for a spectral data set is both symmetric and real. (45) (46)
Chapter 629 Principle Component; PC so (47) (48)
Chapter 630 Fractional information content (49)
Chapter 631 Spectral Decomposition: (51)
Chapter 632 Singular Value Decomposition (SVD): The SVD can be used to decompose a non square matrix into a sum of components expressed as (51)
Chapter 633 If we can solve for the singular values, we can express A as: (52)
Chapter 634 Bases Vectors from SVD: Consider a matrix A operating on a vector x to yield a vector b according to: (53) If A is singular then, there is a subspace of x called the null subspace that will map to zero i.e. (54)