# vm for irm theorem if v1 irmr has orthonormal columns

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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 0nF 6" &'# G \$ Q ¦¡ A \$ " & ' w  (& e    4   0 ¥(¥ I §) G \$ # %¦Q§% # §¤ §    F¨ F &'\$...
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## This document was uploaded on 02/10/2014.

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