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Unformatted text preview: Lecture Note 111: April 2, 2007 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310C Linear Algebra and Its Applications Spring, 2007 Produced by Jeff ChakFu WONG 1 H ERMITIAN M ATRICES HERMITIAN MATRICES 2 • Let C n denote the vector space of all ntuples of complex numbers. • The set C of all complex numbers will be taken as our filed of scalars. • We have already seen that a matrix A will real entries may have complex eigenvalues and eigenvectors . The goal of this lecture is to study matrices with complex entries and look at the complex analogues of symmetric and orthogonal matrices . C OMPLEX I NNER P RODUCTS If α = a + bi is a complex scalar, the length of α is given by  α  = √ αα = p a 2 + b 2 The length of a vector z = ( z 1 ,z 2 , ··· ,z n ) T in C n is given by k z k = (  z 1  2 +  z 2  2 + ··· +  z n  2 ) 1 2 = ( z 1 z 1 + z 2 z 2 + ··· + z n z n ) 1 2 = ( z T z ) 1 2 = ( z · z ) 1 2 As a notational convenience, we write z H for the transpose of z . Thus z T = z H and k z k = ( z H z ) 1 2 Definition: Let V be a vector space over the complex numbers. An inner product on V is an operation that assigns to each pair of vectors z and w in V a complex number h z , w i satisfying the following conditions I. h z , w i ≥ with equality if and only if z = . II. h z , w i = h w , z i for all z and w in V . III. h α z + β w , u i = α h z , u i + β h w , u i . • Note that for a complex inner product space h z , w i = h w , z i , rather than h w , z i . • If we make the proper modifications to allow for this, the theorems on real inner product spaces in Lecture Note 92, will all be valid for complex inner product spaces. COMPLEX INNER PRODUCTS 6 • In particular, let us recall Theorem 0.2 (cf. Lecture Note 81) in term of real inner product spaces; if { u 1 , ··· , u n } is an orthonormal basis for a real inner product space V and x = n X i =1 c i u i then c i = h u i , x i = h x , u i i and k x k 2 = n X i =1 c 2 i • In the case of a complex inner product space , if { w 1 , ··· , w n } is an orthonormal basis and z = n X i =1 c i w i then c i = h z , w i i , c i = h w i , z i and k z k 2 = n X i =1 c i c i We can define an inner product on C n by h z , w i = w H z (1) for all z and w in C n . One easily verifies that Equation (1) actually does define an inner product on...
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 Spring '11
 JeffWong
 Linear Algebra, Algebra, Matrices, Orthogonal matrix, Hilbert space, Normal matrix

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