Linear Algebra with Applications (3rd Edition)

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Algebra and Trigonometry: Real Mathematics, Real People
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Chapter 4 / Exercise 63
Algebra and Trigonometry: Real Mathematics, Real People
Larson
Expert Verified
It follows that A 2 = - 4 3 / 4 - 3 / 4 - 1 / 4 1 / 4 = - 4 P 2 is the required decomposition, where P 2 is a projection matrix.
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Algebra and Trigonometry: Real Mathematics, Real People
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Chapter 4 / Exercise 63
Algebra and Trigonometry: Real Mathematics, Real People
Larson
Expert Verified
8. [10 marks] 8a. [3 marks] Find the lengths and the inner product x · y of the following complex vectors x = 2 - 4 i 4 i , y = 2 4 ( i 2 = - 1) . 8b. [3 marks] Let A = 1 1 - i 1 + i 2 . Let x 1 , x 2 be two (linearly independent) eigen- vectors of A . Compute x 1 · x 2 and show that det( A ) R . 8c. [4 marks] Prove that for any complex vector x x H Ax R . ( H = Hermitian) Sol. [8a.] length( x ) = ( x H x ) 1 / 2 = ( £ 2 + 4 i - 4 i / · 2 - 4 i 4 i ) 1 / 2 = 6; length( y ) = ( y T y ) 1 / 2 = 20 and x · y := x H y = 4(1 - 2 i ). [8b.] Notice that A = A H , furthermore let λ i be the 2 eigenvalues of A : 0 = λ 1 6 = λ 2 = 3, then x 1 · x 2 = 0. Also, one knows that every eigenvalue of a Hermitian matrix is real and so will be its determinant (det( A ) = λ 1 λ 2 = 0). [8c.] We have ( x H Ax ) H = x H Ax , as A = A H . It follows that x H Ax R .

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