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    Ij Ji: Typeset by Ams-Tex

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    salman suhail
    The document discusses normal operators and orthonormal bases of eigenvectors. It defines the adjoint of a linear operator T, denoted T*, and proves that if T and S are represented by matrices using orthonormal bases, then Sij = Tji. It then proves that a linear operator T on a finite-dimensional complex vector space has an orthonormal basis of eigenvectors if and only if T*T = TT*. Finally, it discusses special classes of normal operators, including self-adjoint, skew-adjoint, and unitary operators.

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    Ij Ji: Typeset by Ams-Tex

    Uploaded by

    salman suhail
    21 views3 pages
    The document discusses normal operators and orthonormal bases of eigenvectors. It defines the adjoint of a linear operator T, denoted T*, and proves that if T and S are represented by matrices using orthonormal bases, then Sij = Tji. It then proves that a linear operator T on a finite-dimensional complex vector space has an orthonormal basis of eigenvectors if and only if T*T = TT*. Finally, it discusses special classes of normal operators, including self-adjoint, skew-adjoint, and unitary operators.

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    coherent states

    Original Title

    Normal

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    The document discusses normal operators and orthonormal bases of eigenvectors. It defines the adjoint of a linear operator T, denoted T*, and proves that if T and S are represented by matrices using orthonormal bases, then Sij = Tji. It then proves that a linear operator T on a finite-dimensional complex vector space has an orthonormal basis of eigenvectors if and only if T*T = TT*. Finally, it discusses special classes of normal operators, including self-adjoint, skew-adjoint, and unitary operators.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    21 views3 pages

    Ij Ji: Typeset by Ams-Tex

    Uploaded by

    salman suhail
    The document discusses normal operators and orthonormal bases of eigenvectors. It defines the adjoint of a linear operator T, denoted T*, and proves that if T and S are represented by matrices using orthonormal bases, then Sij = Tji. It then proves that a linear operator T on a finite-dimensional complex vector space has an orthonormal basis of eigenvectors if and only if T*T = TT*. Finally, it discusses special classes of normal operators, including self-adjoint, skew-adjoint, and unitary operators.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    Normal Operators and

    Orthonormal Bases of Eigenvectors


    Let V and W be a finite-dimensional complex Euclidean vector spaces.
    Theorem 1. Let T : V W be a linear map. Then there is a unique linear map
    S : W V such that
    (1)

    (T v, w) = (v, Sw) for all v V and w W .

    If we use orthonormal bases for V and for W to represent T and S by matrices, then
    Sij = Tji .
    Definition. The map S is called the adjoint of T and is denoted T .
    Proof. Let e1 , . . . , en and f1 , . . . , fm be orthonormal bases for V and for W , respectively.
    Then
    X
    X
    (T v, w) = (T (
    vi ei ),
    wj fj )
    X
    =
    vi wj (T ei , fj )
    i,j

    vi wj (the fj component of T ei )

    vi wj Tji .

    i,j

    (2)

    i,j

    Almost exactly the same calculation shows that


    X
    (3)
    (v, Sw) =
    vi wj Sij .
    i,j

    Thus if we let Sij is the conjugate of Tji (for all i and j), then (2) and (3) will be equal
    for all v and w. This establishes the existence of a map S with property (1).
    Conversely, if S has the property (1), then (2) and (3) will be equal for all v and w. In
    particular, they will be equal when v = ei and w = fj , in which case (2) is Tji and (3) is
    the conjugate of Sij . This proves uniqueness. 
    Theorem 2. Let V be a nite-dimensional complex Euclidean vector space. A linear map
    T : V V has an orthonormal basis of eigenvectors if and only if T T = T T .
    Remark. A linear transformation T : V V such that T T = T T is called normal.
    Typeset by AMS-TEX
    1

    Lemma 1. If T : V V is normal, then |T v| = |T v| for all v.


    Proof of lemma 1.
    |T v|2 = (T v, T v) = (T T v, v) = (T T v, v) = (T v, T v) = |T v|2
    
    Lemma 2. Suppose T : V V is normal. If v is an eigenvector of T with eigenvalue ,
    In other words,
    then it is also an eigenvector of T , with eigenvalue .

    T v = v = T v = v.

    Proof. If T is normal, then so is (T I). Thus by lemma 1 applied to T I,

    |(T I)v| = |(T I) v| = |T v v|


    Since T v = v, the left side is 0. Therefore the right side is 0, which means that T v =

    v.
    
    Proof of theorem 2. Any linear transformation of a complex vector space has at least one
    eigenvector. So let v1 be a unit eigenvector for T .
    Now suppose we have orthonormal eigenvectors v1 , . . . , vk for T . If k = dim V , we are
    done. If not, let W be the set of all vectors that are orthogonal to v1 , . . . , vk :
    W = {x V : (x, vi ) = 0 for i = 1, . . . , k}.
    Then W is an n k dimensional subspace (where n = dim V .)
    Claim: If x is perpendicular to vi , then so is T x.
    Proof of claim. If x is perpendicular to vi , then
    (T x, vi ) = (x, T vi ) = (x, i vi ) = (x, vi ) = 0.
    Thus (T x, vi ) = 0, so T x is perpendicular to vi . This proves the claim.
    Consequently, if x is perpendicular to all the vi s, then so is T x. In other words,
    x W = T x W.
    Now since T maps W into itself, W contains a unit eigenvector vk+1 .
    Continuing in this way, we get an orthonormal basis of eigenvectors.

    Special Classes of Normal Transformations


    Note that if T = T , or if T = T , or if T = T 1 , then T is normal. These conditions
    define the following classes of normal operators:
    1. The eigenvalues of a normal operator T are real if and only if T = T . Such a T is
    called self-adjoint or Hermitian.
    2. The eigenvalues of a normal operator T are imaginary if and only if T = T . Such a
    T is called skew-adjoint or skew-Hermitian.
    3. The eigenvalues of a normal operator T are unit complex numbers (i.e., have norms 1)
    if and only if T = T 1 . Such a T is called unitary.
    Unitary operators have other important characterizations:
    Proposition. The T : V V be an operator on the nite dimensional complex Euclidean
    space V . The following are equivalent:
    (1) T is unitary.
    (2) (T u, T v) = (u, v) for all vectors u and v in V .
    (3) |T v| = |v| for all vectors v in V .
    Proof. Exercise.

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