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Lecture 9 Symmetric Matrices Subspaces and Nullspaces

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. 0. R. r.

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Lecture 9 Symmetric Matrices Subspaces and Nullspaces

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  1. Lecture 9Symmetric MatricesSubspaces and Nullspaces Shang-Hua Teng

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

  3. Transpose

  4. Inner Product and Outer Product

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

  6. 0 R r Ellipses and Ellipsoids

  7. 0 r R Later Relating to

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

  9. Symmetric Matrix B is an m by n matrix

  10. Elimination on Symmetric Matrices • If A = AT can be factored into LDU with no row exchange, then U = LT. In other words The symmetric factorization of a symmetric matrix is A = LDLT

  11. 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

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

  13. Properties of Vector Spaces

  14. Other Vector Spaces

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

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

  17. 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

  18. Subspace of R3 • (Z): {(0,0,0)} • (L): any line through (0,0,0) • (P): any plane through (0,0,0) • (R3) the whole space A subspace containing v and w must contain all linear combination cv+dw.

  19. Subspace of Rn • (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 • (Rn) the whole space

  20. Subspace of 2 by 2 matrices

  21. Express Null Space by Linear Combination • A = [1 1 –2]: x + y -2z = 0 x = -y +2z Pivot variable Free variables • 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)}

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

  23. Permute Rows and Continuing Elimination (permute columns)

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

  25. Echelon Matrices Free variables

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