Unformatted text preview: Matrix inverse
Square matrix nxn and general matrix Square mxn Left inverse and right inverse: Left conformity and dimensions of identity matrix Transpose and inverse: similarities and Transpose differences.
35 35 Orthogonal matrices
Forming orthogonal matrix using Forming orthonormal vectors (as rows or as columns?) Orthogonal vectors are perpendicular to Orthogonal each others and must be LI (note GramSchmidt) How to demonstrate that How transpose=inverse (left or right)?
36 36 Matrix and the four vector subspaces
A is an mxn matrix: (a) as n m-dimensional is column vectors (b) as m n-dimensional row vectors Two subspaces in m-dimensional space: Two Column space and Nullspace of AT. Are they orthogonal complements? Any intersection? Two subspaces in n-dimensional space:Row Two space and Nullspace of A. 37 37 Rank and dimensions of vector subspaces
Dimension of Column space = Dimension Dimension of Row space = Rank(A) = r How is r related with m and n? How Consider the m-dimensional space: Consider m=r+ Dimension of Nullspace(AT) Consider the n-dimensional space: n=r+ Consider Dimension of Nullspace(A)
38 38 Full rank (or maximum rank)
Full column rank is n but should be less Full than or equal to m. Why? Full row rank is m but should be less Full than or equal to n. Full rank of a square matrix Full A nxn square matrix A is singular if it is nxn not of full rank n, or Det(A)=0.
39 39 Examples to find subspaces and bases for general A matrix
Cases for m>n and for n>m Cases
• Rank of A (r LI vectors using Gauss elimination) • Span of Column vectors and Nullspace(AT) • Bases for Column space and for Nullspace(AT) • Span of Row vectors and Nullspace(A) • Bases for Row space and for Nullspace(A)
40 Rank and Linear equations
Solutions for Ax=y where A is mxn Solutions The roles of rank: r(A) and r(A | y) The No solution iff r(A) and r(A|y) are No different (Interpretation?) A unique solution iff r(A)=r(A|y)=n unique An (n-r)-parameter family of solutions iff An r(A)=r(A|y) = r is less than n
41 41 Really no solutions?
How should we interpret the case when How m>n (more equations than unknowns) and an left inverse of A exists? Left inverse of A is nxm Left
• Consider (ATA)-1AT and check • How should we interpret x when we get the solutions using left inverse? 42 Unique solutions?
We get a unique solution for x iff We r(A)=r(A|y)=n (or full column rank) Is the mapping Ax=y one-to-one? Is Then how should we find the inverse Then mapping? Are there a “pseudoinverse” for A? 43 43 Rank and Homogeneous equations
Solutions for Ax=0 where A is mxn Solutions The role of rank: r(A) The Always consistent Always Trivial solution x=0 Trivial Unique solution iff r(A)=n Unique An (n-r)-parameter family of nontrivial An solutions iff r(A)=r < n
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- Spring '10
- Linear Algebra, Rank, Invertible matrix