Linear_Operators - EECS 221 A Linear Operators A Notation B...

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Unformatted text preview: EECS 221 A Linear Operators A Notation B Linear Operators C Change of Basis, Range Spaces, and Null Spaces D Eigenvectors and Eigenvalues E Matrix polynomials and functions F Hermitian and Definite Matrices G The Singular-Value Decomposition H Operator Theory 1 A. Notation R ( A ) range space of the operator A N ( A ) null space of the operator A A the transpose of the matrix A A * the adjoint of the operator A , or the complex-conjugate-transpose of the matrix A A > a positive-definite matrix A ≥ a positive-semi-definite matrix λ i ( A ) i th eigenvalue of A Spec( A ) the set of eigenvalues of A ρ ( A ) spectral radius of A = max i | λ i ( A ) | σ i ( A ) i th singular value of A (in descending order) σ largest singular value σ smallest nonzero singular value 2 B. Linear Operators 1 Let V and W be vector spaces over the same base field F . Definition A linear operator is a mapping M : V-→ W such that for all v 1 ,v 2 ∈ V and all α ∈ F (a) M ( v 1 + v 2 ) = M ( v 1 ) + M ( v 2 ) (additivity) (b) M ( αv 1 ) = α M ( v 1 ) (homogeneity) 2 Examples The following operators are linear: ƒ M : C n → C m : v → Av where A ∈ C m × n ƒ M : C (-∞ , ∞ ) → R : f ( t ) → f (0). Is the operator M above familiar ? ƒ Suppose h ( t ) ∈ L 2 [ a,b ] and consider M : L 2 [ a,b ] → R : f ( t ) → Z b a h ( t ) f ( t ) dt Why do we insist that f ∈ L 2 [ a,b ] above ? ƒ Let A,B,X ∈ R n × n and consider M : R n × n → R n × n : X → AX + XB 3 Linear operators on finite dimensional vector spaces are matrices with the action of matrix-vector multiplication. Theorem Consider a linear operator M : V → W where dim ( V ) = n,dim ( W ) = m . Let B = { b 1 , ··· ,b n } and C = { c 1 , ··· ,c n } be basis for V and W respectively. Suppose M ( b j ) = m X i =1 α i,j c i . Let v = ∑ j γ j b j . Then, M ( v ) = m X i =1 τ i c i where τ = τ 1 . . . τ m = α 1 , 1 ··· α 1 ,n . . . ··· . . . α m. 1 ··· α m,n β 1 . . . β n = Mβ The above result says that if we make the canonical (bijective) association of v with β and of M ( v ) with τ , then the action of the operator M corresponds to matrix-vector multiplication as τ = Mβ . The matrix M clearly depends on the particular choice of basis made and is called the matrix representation of M with respect to these basis. 3 Let us explain this result another way. Since V is of dimension n , it is isomorphic to C n . In other words (see Notes on Vector Spaces, B-11 ) , there exists a bijection φ : V → C n such that for all v 1 ,v 2 ∈ V and α ∈ C φ ( v 1 + v 2 ) = φ ( v 1 ) + φ ( v 2 ) , and φ ( αv 1 ) = αφ ( v 1 ) This isomorphism can be exhibited explicity as follows. Fix a basis B = { b 1 , ··· ,b n } for V . Any vector v ∈ V can be expressed as a (unique) linear combination v = n X i =1 α i b i ....
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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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Linear_Operators - EECS 221 A Linear Operators A Notation B...

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