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    UNIT-III

    Real and Complex Matrices, Quadratic Forms

    Real Matrices Symmetric, Skew-symmetric, Orthogonal

    Linear transformations- Orthogonal Transformation

    Complex Matrices- Hermitian, Skew-Hermitian and Unitary

    Eigen Values and Eigen Vectors of Complex matrices and Their Properties

    Quadratic forms- Reduction to Canonical Form

    Rank- Positive, Negative Definite; Semi definite Index, Signature- SylvesterLaw

    Objective Type Questions

    Summary

    1. Definitions:

    Symmetric matrix

    In linear algebra, a symmetric matrix is a square matrix, A, that is equal to its transpose

    The entries of a symmetric matrix are symmetric with respect to themain diagonal (top left tobottom right). So if the entries are written as A = (aij), then

    for all indices i and j. The following 33 matrix is symmetric:

    A matrix is called skew-symmetric or antisymmetric if its transpose is the same as its negative. Thefollowing 33 matrix is skew-symmetric:

    Skew-symmetric matrix

    http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Linear_algebra
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    In linear algebra, a skew-symmetric (orantisymmetric orantimetric[1]) matrix is a square matrix

    A whose transposeis also its negative; that is, it satisfies the equation:

    or in component form, if : for all and

    For example, the following matrix is skew-symmetric:

    Compare this with a symmetric matrix whose transpose is the same as the matrix

    or anorthogonal matrix, the transpose of which is equal to its inverse:

    The following matrix is neither symmetric nor skew-symmetric:

    Every diagonal matrixis symmetric, since all off-diagonal entries are zero. Similarly, each diagonal

    element of a skew-symmetric matrix must be zero, since each is its own negative.

    Orthogonal matrix

    In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns (or

    rows) are orthogonalunit vectors(i.e., orthonormal). Because the columns are unit vectors in

    addition to being orthogonal, some people use the term orthonormal to describe such matrices.

    Equivalently, a matrix Q is orthogonal if its transpose is equal to its inverse:

    alternatively,

    (OR)

    http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Skew-symmetric_matrix#cite_note-0http://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Orthogonal_matrixhttp://en.wikipedia.org/wiki/Orthogonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Orthonormalityhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Skew-symmetric_matrix#cite_note-0http://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Orthogonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Orthonormalityhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Inverse_matrix
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    Definition: An n n matrix A is called an orthogonal matrix whenever T A A I = .

    EXAMPLE:

    1 0 1 0 1 0 cos sin, , ,

    0 1 0 1 0 1 sin cos

    Conjugate transpose

    "Adjoint matrix" redirects here. An adjugate matrixis sometimes called a "classical adjoint matrix".

    In mathematics, the conjugate transpose, Hermitian transpose, oradjoint matrix of an m-by-

    nmatrixA withcomplex entries is the n-by-m matrixA* obtained fromA by taking thetranspose and then

    taking the complex conjugateof each entry (i.e. negating their imaginary parts but not their real parts). The

    conjugate transpose is formally defined by

    where the subscripts denote the i,j-th entry, for 1 i n and 1 j m, and the overbar denotes a

    scalarcomplex conjugate. (The complex conjugate ofa + bi, where a and b are reals, isa bi.)

    This definition can also be written as

    where denotes the transpose and denotes the matrix with complex conjugated entries.

    Other names for the conjugate transpose of a matrix are Hermitian conjugate, ortransjugate. The

    conjugate transpose of a matrixA can be denoted by any of these symbols:

    or , commonly used in linear algebra

    (sometimes pronounced "Adagger"), universally used in quantum mechanics

    , although this symbol is more commonly used for theMoore-Penrose pseudoinverse

    In some contexts, denotes the matrix with complex conjugated entries, and thus the conjugate transpose

    is denoted by or .

    EXAMPLE:

    then

    Hermitian matrix

    http://en.wikipedia.org/wiki/Adjugate_matrixhttp://en.wikipedia.org/wiki/Adjugate_matrixhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Dagger_(typography)http://en.wikipedia.org/wiki/Dagger_(typography)http://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Moore-Penrose_pseudoinversehttp://en.wikipedia.org/wiki/Moore-Penrose_pseudoinversehttp://en.wikipedia.org/wiki/Adjugate_matrixhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Dagger_(typography)http://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse
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    A Hermitian matrix (orself-adjoint matrix) is a square matrixwith complex entries which is equal to its

    own conjugate transpose that is, the element in the ith row andjth column is equal to the complex

    conjugateof the element in thejth row and ith column, for all indices iandj:

    If the conjugate transpose of a matrix is denoted by , then the Hermitian property can be written

    concisely as

    Hermitian matrices can be understood as the complex extension of a real symmetric matrix.

    For example,

    is a Hermitian matrix

    Skew-Hermitian matrix

    In linear algebra, a square matrix with complexentries is said to be skew-Hermitian or

    antihermitian if itsconjugate transposeis equal to its negative.[1] That is, the matrix A is skew-

    Hermitian if it satisfies the relation

    where denotes the conjugate transpose of a matrix. In component form, this means that

    for all i and j, where ai,j is the i,j-th entry ofA, and the overline denotes complex conjugation.

    Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric

    matrices, or as the matrix analogue of the purely imaginary numbers.[2]

    Unitary matrix

    In mathematics, a unitary matrix is an n by ncomplexmatrixUsatisfying the condition

    where is theidentity matrix in n dimensions and is the conjugate transpose (also called the

    Hermitian adjoint) ofU. Note this condition says that a matrix Uis unitary if and only if it has an

    inverse which is equal to its conjugate transpose

    A unitary matrix in which all entries are real is anorthogonal matrix. Just as an orthogonal matrixG preserves the (real)inner product of two real vectors,

    http://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Skew-Hermitian_matrix#cite_note-0http://en.wikipedia.org/wiki/Skew-Hermitian_matrix#cite_note-0http://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Skew-Hermitian_matrix#cite_note-HJ85S412-1http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Hermitian_adjointhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Orthogonal_matrixhttp://en.wikipedia.org/wiki/Orthogonal_matrixhttp://en.wikipedia.org/wiki/Realhttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Vectorshttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Skew-Hermitian_matrix#cite_note-0http://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Skew-Hermitian_matrix#cite_note-HJ85S412-1http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Hermitian_adjointhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Orthogonal_matrixhttp://en.wikipedia.org/wiki/Realhttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Vectors
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    so also a unitary matrix Usatisfies

    for all complex vectors x and y, where stands now for the standard inner producton

    .

    2. Properties of eigen values of real and complex matrices aregiven:

    1.If is a characterstic root of an orthogonal matrix, then 1/ is also a

    characterstic root.

    2.The eigen values of an orthogonal matrix are of unit modulus.

    3. The eigen values of a hermitian matrix are all real.

    4. The eigen values of a real symmetric matrix are all real.

    5. The eigen values of a skew hermitian matrix are either purely

    imaginary or zero.6.The eigen values of a real skew symmetric matrix are purely

    imaginary or zero.

    7. The eigen values of a unitary matrix are of unit modulus.

    8. If A is nilpotent matrix, then 0 is the only eigen value of A

    9. If A is involuntary matrix its possible eigen values are 1 and -1

    10.If A is an idempotent matrix its possible eigen values are 0 and 1

    3. Transformations:

    (a) The transformations X = AY where A = (aij)nXn; X = [x1 x2 . xn];Y = columns of [y1 y2 . yn]; transforms vector Y to vector X over the matrix

    A.

    The transformations is linear.

    (b) Non-singular transformation:

    (i) If A is non-singular, (A 0 ) then Y = AX is non-singular

    transformation.

    (ii) Then, X = A-1Y is inverse transformation of Y = AX.

    (c) Orthogonal transformation: If A is an orthogonal matrix, then

    Y = AX is an orthogonal transformation;

    A is orthogonal , A1 = A- 1 => Y1Y = X1X

    i.e., Y = AX transforms ( x12 + x2

    2 +.+ xn2) to (y1

    2 + y22 +.. +yn

    2)

    4. Quadratic forms: A homogeneous polynomial of 2nd degree in n

    variables x1, x2,xn is called of quadratic form.

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    Thus , q = aijxixj from i , j = 1 to n

    (or) q = [a11x12 + a22x2

    2 ++ annxn2 + (a12+a21)x1x2 +

    (a13+a31)x1x3 ++]

    is a quadratic form in n variables x1,x2xn.

    5. Matrix of a quadratic form q: If A is a symmetric matrix,

    q = X1AX is the matrix representation of q and A is the matrix

    of q where ,(aij+aji)=2aij is coefficient of xixj.[i.e. aij=aji=1/2 coefficient of xixj]

    Then q = X1AX = [x1x2.xn] A columns of[x1 x2.. xn]

    6. Rank of quadratic form: If q = X1AX, then rank of A is the rank

    of quadratic form q

    (a) If rank of A = r = n , q is non-singular form

    (b) If r < n , q is singular

    7. Canonical form or Normal form of q: A real quadratic form q

    in which product terms are missing (i.e. all terms are square terms

    only) is called the canonical form of q.

    i.e. q = a1x12 + a2x2

    2 + + anxn2 is canonical form.

    8. Reduction to canonical form: If D = Diag [d1, d2,.dr] is the

    diagonalization of A, then q1 = d1x12 + d2x2

    2 + . + drxr2 ,

    (where r = rank of A) is canonical form of q = X

    1

    AX.

    9. Nature of a quadratic form:1.If q= X1AX is the given quadratic form (in n variables) of rank r,

    then, q1=d1x12 + d2x2

    2 +.+ drxr2 is the canonical for of q.

    [di is +ve, -ve, or zero]

    (a) Index: The number of +ve terms in q1 is called the index s of

    quadratic form q

    (b) The number of non +ve terms = r-s

    (c) Signature = S- (r-s)= 2s-r.

    2. The quadratic form q is said to be

    (a) +ve definite if r=n, and s=n

    (b) ve definite if r=n, and s=0

    (c) +ve semi-definite if r

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    (e) Indefinite in all other cases

    3. To find the nature of q with the help of Principal minors:

    Let q=X1AX be the given quadratic form and let M1,M2,M3,..

    be the principal minors of A.

    (a) q is +ve definite iff Mj>0 for every jn

    (b) q is ve definite if M1,M3,M5.are all ve and M2,M4,M6,.bethe principal minors of A.

    (c) q is +ve semi-definite if Mj0 for every jn and at least one

    Mj=0.

    (d) q is ve semi-definite if in case (b) some Mj are =0.

    (e) In all other cases q is indefinite

    4. To find the nature of q by examining eigen values of A:-

    If q =X

    1

    AX is quadratic form in n variables then, it is

    a. +ve definite iff all eigen valus of A are +ve.

    b. ve definite iff all eigen values are ve

    c. +ve semi-definite if all eigen values are 0 and at least one eigen value

    =0.

    d. ve semi-definite if all eigen values are 0 and at least one eigen value

    is zero.

    e. Indefinite if A has +ve as well as ve eigen values.

    10. Methods of Reduction of quadratic form to the canonical form.

    (a) Lagranges method: A quadratic form can be reduced by this method to

    a canonical form by completion of squares.

    (b) Diagonalization method: Write A=I3AI3 [if A=(aij)3x3] apply

    elementary row transformation on L.H.S and on prefactor of R.H.S.

    Apply corresponding column transformations on L.H.S as well as the

    post-factor of R.H.S continue this process till the equation is reduced to

    the form,

    D = P1 A P , where D is a diagonal matrix D = [d1 0 0]

    [0 d2 0]

    [0 0 d3]

    Then the canonical form is q1=y1(P1AP)Y=Diag(d1 d2 d3) where

    Y = [y1 y2 y3], i.e., if q = X1 A X, X = [x1 x2 x3] ,

    q1 = d1y12 + d2y2

    2 + d3y32.

    Here X=PY is corresponding transformation.

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