Lecture 9 Symmetric Matrices Subspaces and Nullspaces Shang-Hua Teng

Matrix Transpose Addition: A+B Multiplication: AB Inverse: A -1 Transpose : A T

Transpose

Inner Product and Outer Product

Properties of Transpose End of Page 109: for a transparent proof

Ellipses and Ellipsoids R r 0

Later R r 0 Relating to

Symmetric Matrix Symmetric Matrix: A= A T 1;John 2:Alice 4:Anu 3:Feng Graph of who is friend with whom and its matrix

Examples of Symmetric Matrices B is an m by n matrix

Elimination on Symmetric Matrices If A = A T can be factored into LDU with no row exchange, then U = L T. In other words The symmetric factorization of a symmetric matrix is A = LDL T

So we know Everything about Solving a Linear System Not quite but Almost Need to deal with degeneracy (e.g., when A is singular) Let us examine a bigger issues: Vector Spaces and Subspaces

What Vector Spaces Do We Know So Far R n : the space consists of all column (row) vectors with n components

Properties of Vector Spaces

Other Vector Spaces

Vector Spaces Defined by a Matrix For any m by n matrix A Column Space: Null Space:

General Linear System The system Ax =b is solvable if and only if b is in C(A)

Subspaces A subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: if v and w are vectors in the subspace and c is any scalar, then –v+w is in the subspace –cv is in the subspace

Subspace of R 3 (Z): {(0,0,0)} (L): any line through (0,0,0) (P): any plane through (0,0,0) (R 3 ) the whole space A subspace containing v and w must contain all linear combination cv+dw.

Subspace of R n (Z): {(0,0,…,0)} (L): any line through (0,0,…,0) (P): any plane through (0,0,…,0) … (k-subspace): linear combination of any k independent vectors (R n ) the whole space

Subspace of 2 by 2 matrices

Express Null Space by Linear Combination A = [1 1 –2]: x + y -2z = 0 x = -y +2z Free variables Pivot variable Set free variables to typical values (1,0),(0,1) Solve for pivot variable: (-1,1,0),(2,0,1) {a(-1,1,0)+b(2,0,1)}

Express Null Space by Linear Combination Guassian Elimination for finding the linear combination: find an elimination matrix E such that EA = free pivot

Permute Rows and Continuing Elimination (permute columns)

Theorem If Ax = 0 has more unknown than equations (m > n: more columns than rows), then it has nonzero solutions. There must be free variables.

Echelon Matrices Free variables