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mat 2310 0607_2_note11_1

# mat 2310 0607_2_note11_1 - Lecture Note 11-1 April 2 2007...

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Unformatted text preview: Lecture Note 11-1: April 2, 2007 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310C Linear Algebra and Its Applications Spring, 2007 Produced by Jeff Chak-Fu WONG 1 H ERMITIAN M ATRICES HERMITIAN MATRICES 2 • Let C n denote the vector space of all n-tuples 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 9-2, will all be valid for complex inner product spaces. COMPLEX INNER PRODUCTS 6 • In particular, let us recall Theorem 0.2 (cf. Lecture Note 8-1) 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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mat 2310 0607_2_note11_1 - Lecture Note 11-1 April 2 2007...

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