Unformatted text preview: Matrices
Row and column indexes (i and j) Row Properties similar to vector space as Properties regard to addition and scalar multiplication Matrix multiplication AB=C (note Matrix conformity (being conformable) and element indexes) AB = BA (?) AB
27 27 Square, order, diagonal, identity
Nilpotent, upper and lower triangular Nilpotent, Transpose of a matrix product (proof Transpose and note the contrast with AB) Symmetric and Skew-symmetric Symmetric Square = Symmetric + Skew-symmetric Square (proof) Forming Symmetric matrix using AAT Forming
28 LU Decomposition (factorization of a square matrix)
A = LU LU Using Gauss elimination to factorize Using
• Note how L can be formed from the “factors” used in obtaining U Solving Ax=y by considering Lv=y and Solving Ux=v
• Forward and backward substitution • Any advantages over Gauss elimination?
29 29 There are several LU forms
Gauss elimination – DooLittle Gauss decomposition with L’s diagonal elements all ones Crout decomposition is using a U with Crout diagonal elements all ones Cholesky factorization: An interesting Cholesky factorization for symmetric matrix A = LLT
30 Determinants (for square matrix A only)
Notation: det A or | A | and it is a scalar Notation:
• |A| may have confusion with absolute value if A is a scalar (a 1x1 matrix) Evaluation of det A from definition (Cofactor Evaluation and minor expansions) is computationally expensive for large n ( in n! time) Det is not linear but Det(AB)=DetA DetB Det Det for L and U are simple Det 31 31 Finding det(A)
Elementary row (or column) operations Elementary have interesting effects on det A Using Gauss-elimination like procedure Using to find det A is computationally effective (in n3 time) Examples (simple tricks and Examples observations always help)
32 Matrix rank r(A)
It is a scalar defined for a general matrix It (not necessarily square) It is related to determinant (another It scalar defined only for square matrix) using the concept of submatrices It is also found to relate with LI of a set It of vectors and the solutions of linear equations
33 33 Finding r(A)
If A is mxn, what are the square submatrices If of A?
• Determine the largest order r_max = min(m, n) min(m, • Consider all submatrices of order r, where r= r_max, …, 1 r_max, • Find the largest r where there exists at least one r x r submatrix having non-zero determinant non- A zero matrix has rank 0. zero 34 ...
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