Positive Definite, Semidefinite
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Mathematics/Linear Algebra
Positive Definite, Semidefinite Definition 1. Let TL(V)TL(V) where VV is a finite-dimensional inner product space, and let AMn×n(F)AMn×n(F). Then TT is called positive definite [positive semidefinite] if TT is hermitian and T(x),x>0T(x),x>0 [T(x),x0],x0[T(x),x0],x0, and AA is called positive definite [positive semidefinit..
Lagrange Interpolation Formula
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Mathematics/Linear Algebra
Lagrange Interpolation FormulaLagrange Interpolation Formula, 즉 라그랑주 보간법이라고 불리는 이 방법은 주어진 (n+1)(n+1)개의 점들을 모두 지나는 nn차 이하의 다항식을 유일하게 결정하는 방법이다. Lagrange PolynomialDefinition 1. Let c0,,cnc0,,cn be distinct scalars in an infinite field FF. The lagrange polynomials f0,,fnf0,,fn is defined by $$f_i(x) = \prod_{0 \leq k \neq i \leq n} \frac{x - c_k}{c_i - c_k} \text{ for } 0 \leq i \leq n...
Spectral Theorem
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Mathematics/Linear Algebra
Spectral Theorem Theorem 1. Let TL(V)TL(V) where VV is a finite-dimensional inner product space over FF with the distinct eigenvalues λ1,,λkλ1,,λk. Assume that TT is normal if F=CF=C and that TT is hermitian if F=RF=R. For each i(1ik)i(1ik), let WiWi be the eigenspace of TT corresponding to the eigenvalue λiλi, and that $T_i..
Orthogonal Projection
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Mathematics/Linear Algebra
Orthogonal Projection Definition 1. Let TL(V)TL(V) be a projection, where VV is an inner product space. We say that TT is an orthogonal projection if R(T)=N(T)R(T)=N(T) and N(T)=R(T)N(T)=R(T). Remark Remark. (a) If VV is finite-dimensional, by Theorem 2, we need only assume that one of the preceding conditions holds. (b) If WVWV is finite-dimensional, then there exists..
Projection
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Mathematics/Linear Algebra
Projection Definition 1. Let W1,,WkVW1,,WkV such that V=ki=1WiV=ki=1Wi, and let TL(V)TL(V) where VV is a vector space. Then TT is the projection on WjWj if, whenever x=x1++xkx=x1++xk with xiWi(i=1,,k)xiWi(i=1,,k), we have T(x)=xjT(x)=xj. Theorem 1 Theorem 1. Let W1,,WkVW1,,WkV such that V=ki=1WiV=ki=1Wi where VV is a vecto..
Hermitian Matrix
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Mathematics/Linear Algebra
Hermitian MatrixDefintion 1. Let AMn×n(F)AMn×n(F). We say that AA is hermitian (or self-adjoint) if A=AA=A.선형 연산자가 hermitian일 조건과 동일하게 hermitian인 행렬을 정의할 수 있다. 또한 Theorem 1의 행렬 버전을 말할 수 있다.Theorem 1Theorem 1. Let AMn×n(R)AMn×n(R). Then AA is hermitian AA is orthogonally equivalent to a real diagonal matrix. Proof. The proof is similar to the proof of Theor..
Normal Matrix
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Mathematics/Linear Algebra
Normal MatrixDefintion 1. Let AMn×n(F)AMn×n(F). We say that AA is normal if AA=AAAA=AA.선형 연산자 TT가 normal일 조건과 동일하게 normal인 행렬을 정의할 수 있다. 또한 Theorem 2의 행렬 버전을 말할 수 있다.Theorem 1Theorem 1. Let AMn×n(C)AMn×n(C). Then AA is normal AA is unitarily equivalent to a diagonal matrix.Proof. ()By Theorem 2, there exists an orthonormal basis for VV c..
Unitary, Orthogonal Matrix
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Mathematics/Linear Algebra
Unitary, Orthogonal MatrixDefinition 1. Let AMn×n(F)AMn×n(F). Then AA is called a unitary matrix if AA=AA=IAA=AA=I and is called an orthogonal matrix if AtA=AAt=IAtA=AAt=I.Theorem 1에 근거해 unitary 혹은 orthogonal 행렬의 정의를 위와 같이 할 수 있다.RemarkRemark. AA=IAA=I [AA=IAA=I] the rows [columns] of AA form an orthonormal basis for FnFn.() $\delta_{ij} = I_{Ij} = (AA^*)_{ij} ..
Unitarily, Orthogonally Equivalent
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Mathematics/Linear Algebra
Unitarily, Orthogonally Equivalent Definition 1. Let A,BMn×n(C)A,BMn×n(C) [Mn×n(R)Mn×n(R)]. Then AA and BB are unitarily equivalent [orthogonally equivalent] if there exists a unitary [orthogonal] matrix PP such that A=PBPA=PBP [A=PtBPA=PtBP].
Unitary, Orthogonal Operator
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Mathematics/Linear Algebra
Unitary, OrthogonalDefinition 1. Let TL(V)TL(V) where VV is a finite-dimensional inner product space over FF. If ||T(x)||=||x||,xV||T(x)||=||x||,xV, we call TT a unitary operator if F=CF=C and call TT an orthogonal operator if F=RF=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 ..