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0 dnn ωn Since the matrix D∞ Q∞ commutes with A∞ , and A∞ is unitarily equivalent to A (follows from
the QR algorithm), D∞ Q∞ is unitarily equivalent to A. So λj = djj ωj are the eigenvalues of A.
Thus, we know djj  = λj . Moreover, if the diagonal entries of D∞ are distinct (magnitudes of
the eigenvalues of A are distinct), and since D∞ Q∞ = Q∞ D∞ , the oﬀ diagonal entries of Q∞
must be zeros, and we precisely have Q∞ = Q∞ . Thus, Q∞ is diagonal and unitary, and d11 ω1
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d22 ω2 . .
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A∞ = .
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0 dnn ωn
2 So if the magnitudes of a normal matrix A are distinct and if the QR algorithm converges, then
it diagonalizes A. 1 Real Matrices (cont’d) We proceed to show under which conditions a matrix with entries over R can be diagonalized the
same way as matrices with complex entries.
1.5.1 Deﬁnition. A matrix A ∈ Mn (R) is similar to B ∈ Mn (R) if there exists invertible
S ∈ Mn (R) such that
B = S −1 AS.
The matrix A is diagonalizable if A is similar to a diagonal matrix.
1.5.2 Theorem. A ∈ Mn (R) is diagonalizable if and only if there is a set of n linearly independent
eigenvectors.
Proof. As before, S −1 AS = D, with D diagonal implies that S = [x1 , x2 , · · · , xn ] contains a
basis of n eigenvectors and vice versa.
1.5.3 Theo...
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This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.
 Fall '12
 BernhardBodmann
 Addition, Matrices

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