At many points in the course, we have made assumptions that various matrices or eigenvalues are real while discussing various theorems. If you have noticed, this has always happened in contexts where we have invoked orthononnality during the proof or statement of the result. Now that you understand the idea of orthonormality for complex vectors, and how to adapt GramwSchmitt to complex vectors, you can go back and remove those restrictions. This problem asks you to do exactly that. Unlike many of the problems that we have given you in lot-"t and 1&3, this problem has far less hand- holding _ there aren't multiple parts breaking down each step for you. Fortunately, you have the existing proofs in your notes to work based on. So this problem has a secondary function to help you solidify your understanding of these earlier concepts ahead of the nal exam. 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 9 : ST is what is called a Hermitian matrix H that satises H = H * where H" = F is the mnjugatertranspose of H. [a] The upper~triangularization theorem for all [potentially complex) square matrioes A says that there 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 matrix has an {potentially complex) eigenvalueieigenvector pair. You don't need to prove this. But you can make no other assumptions. (HINT Use the exact same argument .as hefore, just use conjugate-transporter instead oftmmposes.) Congratulations, once you have completed this part you essentially can solve all systems of linear differential equations based on what you know, and you can also complete the proof that having all the eigenvalues beng stable implies BIBC} stability. {b} 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 shouirijust ieverage upper-trianguiarization and use the fact that {ABC 11* : C*B*A*. There is a reason that this part comes after the rst part.) [c] The SVD for complex matrices says that any rectangular {potentially complex} matrix A can be written as A = 2,; timeri: where a,- are real positive numbers and the collection {E} are orthonormal (but potentially complex} as well as {rift}. Adapt the derivation of the SVD from the real ease to prove this theorem. Feel free to assume that A is wide. (Since you can just conjugate-transpose everything to get a tall matrix to become wide.) (HINT Anaiogonsiy to before, you 're going to have to show that the matrix EPA is Hermitian antiI that it has non-negative eigenvaiues. Use the previous part. There is a reason that this part comes after the previous pans. J
