Chapter 1 and 2

Chapter 1 and 2 - Chapter 1 Introduction and Review 1. If A...

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Introduction and Review 1. If A IR n × n and α is a scalar, what is det( αA )? What is det( - A )? Answer 1.1 Suppose A IR n × n and α IR . Denote the identity of IR n × n by I n . Then det( αA ) = det( α ( I n A )) = det(( αI n ) A )= det( αI n ) det A = α n det A . Thus det( - A ) = det(( - 1) A ) = ( - 1) n det A . 2. If A is orthogonal, what is det A ? If A is unitary, what is det A ? Answer 1.2 A is orthogonal = AA T = I = 1 = det I = det( AA T )= det A det A T = det A det A = (det A ) 2 = det A = ± 1 . A is unitary = AA H = I = 1 = det I = det( AA H ) = det A det A = ( re )( re - )= r 2 for some θ, r IR = r = ± 1 , and so det A = e for some θ IR . 3. Let x, y IR n . Show that det( I - xy T ) = 1 - y T x . Answer 1.3 Using determinant properties 16 and 17, det( I n - xy T )= det(1) det( I n - xy T ) = det ± I n x y T 1 ² = det( I n ) det(1 - y T x ) = 1 - y T x . 1
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This note was uploaded on 04/18/2011 for the course EE 205A taught by Professor Laub during the Fall '10 term at UCLA.

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Chapter 1 and 2 - Chapter 1 Introduction and Review 1. If A...

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