Lecture 20, 18.300, Tue Apr 19, 2022.
Summary
Finish Gas Dynamics, Acoustics, and Strings.
Start normal modes. #068---#072
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Review of linear algebra:
1- If A is self-adjoint then it has an orthonormal basis of eigenvectors,
   and its eigenvalues are all real.
   Proof: that the eigenvalues are real is standard. For the rest show that,
   given an eigenvector v, A keeps v^\perp invariant. Hence the dimension
   of the problem can be reduced by one. Then iterate.
2- If A and B are self-adjoint and they commute, then they have a common
   basis of orthonormal eigenvectors.
   Proof: for any eigenvalue l of A, let E_l be the set of corresponding
   eigenvectors [A is diagonal on E_l]. Then B keeps E_l invariant, so a
   base of orthonormal eigenvectors of B can be selected within E_l.
3- Normal matrices have orthonormal basis of eigenvectors.
   Proof: N normal means that it commutes with its adjoint. Then we write
   N = A + i*B, where A = (N+N^*)/2 and B = (N-N^*)/(2*i), and apply 2.

Examples of Normal Matrices:
a- Self-adjoint [real eigenvalues]. ........... A^* = A.
b- Skew-adjoint [imaginary eigenvalues]. ...... A^* = -A.
c- Unitary [eigenvalues in unit circle]. ...... U^* = U^{-1}.
d- Any matrix such that A^* = f(A) --- a function of A;
   (a--c) are special cases of this.
   Then the eigenvalues must satisfy: lambda^* = f(lambda), so they
   are restricted to some curve in the complex plane.

#068 Separation of variables <------------- Students must read on their own.
     See the notes posted on the web page!
#069 Normal modes.
     Special case of self-adjoint operators.
     Give examples.
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