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L! 1 There is one concept that you will need beyond the idea of what orthogonality means for complex vectors as well as the idea

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L! 1 There is one concept that you will need beyond the idea of what orthogonality means for complex vectors as well as the idea of conjugatetransposes of vectors and matrices. The analogy of a real symmetric matrix S that satises S : ST is what is called a Hermitian matrix H that satises H : H\" where h" : F is the conjugatevtranspose of H. {a} The upper-triangulaiization theorem for all (potentially complex] square matrices A says that there lib] {C} exists an orthonormal (possibly complex) basis V so that V'AV is upper-triangular. Adapt the proof from the real ease with assumed real eigenvalues to prove this theorem. Feel free to assume that any square mauix has an (potentially complex) eigenvalueteigenvector pair. You don't need to prove this. But you can make no other aSsumptions. {HINT Use the exact same argument as before, just use conjugate-transporter; instead oftransposes.) Congratulations, once you have completed this part you essentially can solve all systems of linear diflerential equations based on what you know. and you can also complete the proof that having all the eigenvalues being stable implies BIBD stability. The spectral theorem for Hermitian matrices says that a Hermitian matrix has all real eigenvalues and an orthonormal set of eigenvectors. Adapt the proof from the real symmetric case to prove this theorem. (HINT? As before, you should just leverage upper-triangular'ization and use the fact that (AMER?)t 2 Ct 314*. There is a reason that this part comes after the rst part. } The SUD for complex matrices says that any rectangular {mtentially complex] matrix A can be written as A = Efgy-ti?\" where 0'.- are real positive numbers and the collection {if} are orthonormal (but potentially complex] as well as {of}. Adapt the derivation of the 51th from the real ease to prove this theorem. Feel free to assume that A is wide. (Since you can just wnjugatewlranspose everything to get a tall matrix to become wide.) (HINT Anatogausty to before. you're going to have to show that the matrix AV; is Herrnitian and that it has non-negative eigenvalues. Use the previous part. There is a reason that this part comes after the previous parts. )

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