(to the tune of “*If You’re Happy and You Know It*“)

Note: This song describes what to do for a similarity transformation to find the eigenvalues of a matrix A. A similarity transform changes the matrix but does not alter its eigenvalues. Ideally, if you could find a similar matrix that was diagonal, you could pick the eigenvalues right off the diagonal. A similarity transform is of this form: B = T^{-1} A T, where T is the similarity transform matrix, and B is the matrix that is similar to A.

If your matrix has distinct eigenvalues,

It is very much apparent what to use:

Choose nonsingular for T,

and get a diagonal B,

If your matrix has distinct eigenvalues.

If you have a real symmetric matrix A,

Choose orthogonal for T: it’s the best way!

Because, after all

B is real diagonal

If you have a real symmetric matrix A.

If it’s complex and Hermitian, never fear!

The solution to your problem is quite clear:

Use a unitary T,

to get a real diagonal B

If it’s complex and Hermitian, never fear!

If your matrix A is normal as can be,

Take the following advice from me:

If your T is unitary

B’s diagonal (how scary!)

If your matrix A is normal as can be.

If A is chosen arbitrarily,

and you find a T that’s unitary,

Then B has the form of Schur

(that’s upper triangular)

If A is chosen arbitrarily.

If A is once again arbitrary,

Then you choose “nonsingular” for T,

B is almost diagonal,

(that’s Jordan form, y’all!)

If A is once again arbitrary.

“The Similarity Transform Song” Copyright (c) 2000 Rebecca Hartman-Baker.(from

http://www.cse.uiuc.edu/~rjhartma/eigensong.html)

**Found this while surfing. Was really funny and interesting.**

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