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

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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
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A. Notation R ( A ) range space of the operator A N ( A ) null space of the operator A A 0 the transpose of the matrix A A * the adjoint of the operator A , or the complex-conjugate-transpose of the matrix A A > 0 a positive-definite matrix A 0 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
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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 = 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 = . 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
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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 . With the abstract vector v V , we associate the concrete vector φ ( v ) = α 1 .
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