Unitarily, Orthogonally Equivalent
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Mathematics/Linear Algebra
Unitarily, Orthogonally Equivalent Definition 1. Let A,BMn×n(C) [Mn×n(R)]. Then A and B are unitarily equivalent [orthogonally equivalent] if there exists a unitary [orthogonal] matrix P such that A=PBP [A=PtBP].
Unitary, Orthogonal Operator
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Mathematics/Linear Algebra
Unitary, OrthogonalDefinition 1. Let TL(V) where V is a finite-dimensional inner product space over F. If ||T(x)||=||x||,xV, we call T a unitary operator if F=C and call T an orthogonal operator if F=R.유한차원의 경우 unitary, 혹은 othogonal, 즉 유니터리 혹은 직교 연산자라고 부르며, 무한차원의 경우 metric을 보존한다는 점을 강조하기 위해 isometry라고 부른다.     자명하게 선형 연산자 T가 ..
Metric
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Mathematics/Real analysis
Metric Definition 1. Given a nonempty set X, metric is a function d:X×X[0,) such that x,y,zX, the following hold: (a) d(x,y)=0x=y (b) d(x,y)=d(y,x) (c) d(x,z)d(x,y)+d(y,z) Then (X,d) is called a metric space. 위와 같은 성질을 만족하는 함수 d를 metric, 즉 거리라고 부른다.
Hermitian Operator
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Mathematics/Linear Algebra
HermitianDefintion 1. Let TL(V) where V is an inner product space. We say that T is hermitian (or self-adjoint) if T=T.위와 같은 조건을 만족시켰을 때 선형 연산자가 hermitian이라고 부른다. 자명하게 선형 연산자 T가 hermitian일 조건은 [T]β가 hermitian일 조건과 동치이다. (β는 orthonormal basis)    선형 연산자가 normal일 조건을 생각해본다면, hermitian이면 normal임을 쉽게 알 수 있다.LemmaLemma. Let T be a hermitian operator on a ..
Normal Operator
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Mathematics/Linear Algebra
Normal OperatorDefintion 1. Let TL(V) where V is an inner product space. We say that T is normal if TT=TT.위와 같은 조건을 만족시켰을 때 선형 연산자가 normal, 즉 정규하다고 부른다. 자명하게 선형 연산자 T가 normal일 조건은 [T]β가 normal일 조건과 동치이다. (β는 orthonormal basis) Theorem 1Theorem 1. Let T be a normal operator on V where V is an inner product space. Then the following statements ar..
Schur's Theorem
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Mathematics/Linear Algebra
Schur's TheoremTheorem 1. Let TL(V) where V is a finite-dimensional inner product space. Then there exists an orthonormal basis β for V such that [T]β is upper triangular. Proof. Let n=dim(V). The proof is by the mathematical induction on n. If n=1, the result is immediate. So suppose that the theorem is true for n1 where n11. Let W be..
Adjoint of Linear Transformation
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Mathematics/Linear Algebra
행렬의 adjoint는 원 행렬의 켤레 전치로 정의되었다. 유사하게 선형 변환의 adjoint를 정의하려고 한다. 어떤 선형 변환 T에 대해 ([T]βγ)=[U]γβ를 만족하는 선형 변환 U를 찾고, 그 UT의 adjoint라고 정의하는 것이 자연스러울 것이다.Adjoint of Linear TransformationDefinition 1. Let TL(V,W) where V and W are finite-dimensional inner product space with inner products ,1 and $\langle \c..
Bessel's Inequality, and Parseval's Identity
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Mathematics/Linear Algebra
Bessel's Inequality Theorem 1. Let (V,,) be an inner product space, and let S={v1,...,vn} be an orthonormal subset of V. Then xV, ||x||2i=1n|x,vi|2. Proof. Let S=W. Then !yW,zW such that x=y+z by Theorem 1. Thus we have $$||x||^2 = ||y||^2 + ||z|..
Direct Sum
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Mathematics/Linear Algebra
SumDefinition 1. Let W1,...,WkV. We define the sum of these subspaces to be the set {v1++vk|viWi for 1ik}, which we denote by i=1kWi.Direct SumDefinition 2. Let W1,...,WkV. We call V the direct sum of W1,...,Wk and write V=i=1kWi, if V=i=1kWi and $W_j \cap \sum_{i \n..
Orthogonal Complement
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Mathematics/Linear Algebra
Orthogonal ComplementDefinition 1. Let (V,,) be an inner product space, and let SV. We define S to be S={xV|x,y=0,yS}. The set S is called the orthogonal complement of S.S의 벡터들에 직교하는 벡터들을 모두 모아놓은 집합을 S의 orthogonal complement, 직교여공간이라고 부른다. 자명하게 $S^{\perp} \..