vm for irm theorem if v1 irmr has orthonormal columns

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ull orthonormal basis {v1 , . . . , vm } for IRm Theorem If V1 ∈ IRm×r has orthonormal columns, then there exists V2 ∈ IRm×(m−r ) such that V = [V1 V2 ] ∈ IRm×m is orthogonal. Note that ran(V1 )⊥ = ran(V2 ) Proof. This is a standard result from introductory linear algebra 6 / 17 Norms and orthogonal transformations The vector 2-norm is invariant under orthogonal transformation Q Qx 2 2 = x Q Qx = x x = x 2 Likewise, matrix 2-norm and Frobenius norm are invariant with respect to orthogonal transformations Q and Z QAZ QAZ F 2 = = A A F 2 7 / 17 8 / 17 % 3 42 0nF 6" &'# G $ Q ¦¡ A $ " & ' w  (& e    4   0 ¥(¥ I §) G $ # %¦Q§% # §¤ §    F¨ F &'$...
View Full Document

This document was uploaded on 02/10/2014.

Ask a homework question - tutors are online