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Linear Operators and Their Matrices
APPENDIX
LINEAR OPERATORS AND THEIR MATRICES
C
The fundamental objects used in linear algebra are vectors, defined in some three dimensional space with basis functions dx , dy , dz . Definition 48. A linear transformation of a vector is evoked by a rule f which assigns the vector B on the basis of an arbitrary vector, A: B = f (A), and this rule possesses the property of linearity if the following identity applies: (1) (2) f λ1 A + λ2 A = λ1 f A(1) + λ2 f A(2) , where A(1) , A(2) are arbitrary vectors and λ1 , λ2 are arbitrary scalars. The rule evoking a linear transformation of a vector is often simply called a linear operator, f . All linear operators evoke the linear transformation of a vector. Decomposing the vectors A and B onto the basis vectors dx , dy , dz , we have: A = Ax dx + Ay dy + Az dz = Aβ dβ , B = Bx dx + By dy + Bz dz =
β=x,y,z
Bα dα .
α=x,y,z
Making use of the property of linearity for the operator f, we obtain
Bα dα = f Aβ dβ = Aβ f dβ . α=x,y,z
β=x,y,z
β=x,y,z
Furthermore, applying the operator f to the orthogonal basis vectors dx , dy , dz , we obtain some vector fβ which can be decomposed with respect to the basis vectors:
fβ = f dβ = fxβ dx + fyβ dy + fzβ dz = fαβ dα , α=x,y,z
where β = x, y, z. We can now write α=x,y,z
Bα dα =
fαβ Aβ dα .
α=x,y,z β=x,y,z
Foundations of Geophysical Electromagnetic Theory and Methods. DOI: 10.1016/B978-0-44-463890-8.00031-1 Copyright © 2018 Elsevier B.V. All rights reserved.
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APPENDIX C LINEAR OPERATORS AND THEIR MATRICES
Equating the coefficients of the unit orthogonal vectors done by one, we have fαβ Aβ . Bα = β=x,y,z
Clearly, the coefficients fαβ (α = x, y, z; β = x, y, z) form a 3 by 3 matrix: ⎡ ⎤ fxx fxy fxz fαβ = ⎣ fyx fyy fyz ⎦ , fzx fzy fzz which is called the linear operator matrix, f, on the given basis function. Therefore, in the language of matrices, this last equation can be written as: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Bx fxx fxy fxz Ax ⎣ By ⎦ = ⎣ fyx fyy fyz ⎦ · ⎣ Ay ⎦ Bz fzx fzy fzz Az and the components are Bx = fxx Ax + fxy Ay + fxz Az , By = fyx Ax + fyy Ay + fyz Az , Bz = fzx Ax + fzy Ay + fzz Az . Note that, if the vectors A and B that we are considering lie only in the horizontal plane XOY , the matrix form of writing the linear operator f is as follows: Bx fxx fxy Ax = · By fyx fyy Ay and accordingly Bx = fxx Ax + fxy Ay , By = fyx Ax + fyy Ay . We need still another concept from linear algebra. Definition 49. The linear transformation evoked by the operator f is called non-degenerate if the determinant of the linear transformation matrix is not zero: Det fαβ = 0 Definition 50. If this is not true, the linear transformation is called degenerate. Non-degenerate transformation evokes a reversible transformation of the vector A to the vector B. This means that an inverse linear operator, f −1 , exists which permits the inversion of the vector B to the vector A: f −1 (B) = f −1 f (A) .
LINEAR OPERATORS AND THEIR MATRICES
739
An operator f which causes a non-degenerate transformation is called an invertible operator. Operators which result in degenerate transformations are called non-invertible; that is, no inverse operators exist in such cases.
LINEAR OPERATORS AND THEIR MATRICES
C
The fundamental objects used in linear algebra are vectors, defined in some three dimensional space with basis functions dx , dy , dz . Definition 48. A linear transformation of a vector is evoked by a rule f which assigns the vector B on the basis of an arbitrary vector, A: B = f (A), and this rule possesses the property of linearity if the following identity applies: (1) (2) f λ1 A + λ2 A = λ1 f A(1) + λ2 f A(2) , where A(1) , A(2) are arbitrary vectors and λ1 , λ2 are arbitrary scalars. The rule evoking a linear transformation of a vector is often simply called a linear operator, f . All linear operators evoke the linear transformation of a vector. Decomposing the vectors A and B onto the basis vectors dx , dy , dz , we have: A = Ax dx + Ay dy + Az dz = Aβ dβ , B = Bx dx + By dy + Bz dz =
β=x,y,z
Bα dα .
α=x,y,z
Making use of the property of linearity for the operator f, we obtain
Bα dα = f Aβ dβ = Aβ f dβ . α=x,y,z
β=x,y,z
β=x,y,z
Furthermore, applying the operator f to the orthogonal basis vectors dx , dy , dz , we obtain some vector fβ which can be decomposed with respect to the basis vectors:
fβ = f dβ = fxβ dx + fyβ dy + fzβ dz = fαβ dα , α=x,y,z
where β = x, y, z. We can now write α=x,y,z
Bα dα =
fαβ Aβ dα .
α=x,y,z β=x,y,z
Foundations of Geophysical Electromagnetic Theory and Methods. DOI: 10.1016/B978-0-44-463890-8.00031-1 Copyright © 2018 Elsevier B.V. All rights reserved.
737
738
APPENDIX C LINEAR OPERATORS AND THEIR MATRICES
Equating the coefficients of the unit orthogonal vectors done by one, we have fαβ Aβ . Bα = β=x,y,z
Clearly, the coefficients fαβ (α = x, y, z; β = x, y, z) form a 3 by 3 matrix: ⎡ ⎤ fxx fxy fxz fαβ = ⎣ fyx fyy fyz ⎦ , fzx fzy fzz which is called the linear operator matrix, f, on the given basis function. Therefore, in the language of matrices, this last equation can be written as: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Bx fxx fxy fxz Ax ⎣ By ⎦ = ⎣ fyx fyy fyz ⎦ · ⎣ Ay ⎦ Bz fzx fzy fzz Az and the components are Bx = fxx Ax + fxy Ay + fxz Az , By = fyx Ax + fyy Ay + fyz Az , Bz = fzx Ax + fzy Ay + fzz Az . Note that, if the vectors A and B that we are considering lie only in the horizontal plane XOY , the matrix form of writing the linear operator f is as follows: Bx fxx fxy Ax = · By fyx fyy Ay and accordingly Bx = fxx Ax + fxy Ay , By = fyx Ax + fyy Ay . We need still another concept from linear algebra. Definition 49. The linear transformation evoked by the operator f is called non-degenerate if the determinant of the linear transformation matrix is not zero: Det fαβ = 0 Definition 50. If this is not true, the linear transformation is called degenerate. Non-degenerate transformation evokes a reversible transformation of the vector A to the vector B. This means that an inverse linear operator, f −1 , exists which permits the inversion of the vector B to the vector A: f −1 (B) = f −1 f (A) .
LINEAR OPERATORS AND THEIR MATRICES
739
An operator f which causes a non-degenerate transformation is called an invertible operator. Operators which result in degenerate transformations are called non-invertible; that is, no inverse operators exist in such cases.