{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter 1 and 2

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 1 Introduction and Review 1. If A IR n × n and α is a scalar, what is det( α A )? 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( α A ) = det( α ( I n A )) = det(( α I n ) A ) = det( α I 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 j θ )( re - j θ ) = r 2 for some θ , r IR = r = ± 1 , and so det A = e j θ 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 = det( I n ) det(1 - y T x ) = 1 - y T x . 1
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 CHAPTER 1. INTRODUCTION AND REVIEW 4. Let U 1 , U 2 , . . . , U k IR n × n be orthogonal matrices. Show that the product U = U 1 U 2 · · · U k is an orthogonal matrix.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern