Lecture #7 Notes

n must be eigenvalues for b now we are in a position

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: λ1 ∗ · · · ∗ . = 0 , . . 0 B 2 ￿ where B ∈ Mn−1 . Now, the characteristic polynomial of A factors as: PA ( t ) = ( t − λ1 ) PB ( t ) , where PB (t) is the characteristic polynomial of B , which means {λ2 , λ3 , ..., λn } must be eigenvalues for B . Now we are in a position to invoke the induction assumption: there exists U ￿ ∈ Mn−1 such that U ￿ (U ￿ )∗ = I and (U ￿ )∗ BU ￿ = T ￿ , where T ￿ ∈ Mn−1 is upper triangular with λ2 , λ3 , ..., λn on the diagonal . Now define U2 as: 1 0···0 . U2 = 0 , . . 0 U￿ and define U as U = U1 U2 , which is unitary since products of unitary matriecs are unitary. Then: ∗∗ ∗ ∗ U ∗ AU = (U1 U2 )∗ A(U1 U2 ) = U2 U1 AU1 U2 = U2 (U1 AU1 )U2 λ1 ∗··...
View Full Document

Ask a homework question - tutors are online