Lecture #7 Notes

# Well begin by nding a condition for unitary

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Unformatted text preview: ·∗ λ1 ∗ · · · ∗ 0 0 ∗ = U2 . U2 = . . . . . 0 0 λ1 ∗ · · · ∗ . = 0 = T, . . 0 ￿)∗BU ￿ (U B T￿ which is upper triangular with λ1 , λ2 , ..., λn along its diagonal. This completes the proof, since A = (U ∗ ) ∗ T U ∗ . 2.2 Unitary diagonalization for normal matrices Now we will examine the consequences of this theorem. We’ll begin by ﬁnding a condition for unitary diagonalizability. 2.2.2 Deﬁnition. A matrix A ∈ Mn is called normal if: AA∗ = A∗ A . 2.2.3 Remark. Hermitian and unitary matrices are normal, but there are normal matrices which are neither Hermitian nor unitary. ￿ ￿ 1 −1 2.2.4 Example. A = is normal, but it is neither Hermitian nor unitary. 11 2.2.5 Proposition. A matrix A ∈ Mn is normal iﬀ ||Ax|| = ||A∗ x||, for all x ∈ Cn . 3 Proof. First note that: A∗ A = AA∗ ⇔ A∗ A − AA∗ = 0 ⇔ (A∗ A − AA∗ )...
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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.

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