Similarity transformation, im really confused

[TheIsingGuy]

I have been taught two version of the Similarity tranformations on my course, one is from Hamiltonian mechanics, the other is from group theory, I understand neither, all I know is it changes basis, but what can I use it for? I would really appreciate if someone can explain it to me. Thanks

 

[Johan de Vries]

I have been taught two version of the Similarity tranformations on my course, one is from Hamiltonian mechanics, the other is from group theory, I understand neither, all I know is it changes basis, but what can I use it for? I would really appreciate if someone can explain it to me. Thanks

You can use it to compute determinants of infinite matrices of the form A_{i,j} = f(i-j).

The eigenvalue equation is then just a convolution product, which factorizes if you take the Fourier Transform, w.r.t. i and j.

 

[Andy Resnick]

My only exposure to similarity transformations has been in the context of fluid flow; to be honest, I never fully understood it, either.

In fluid flow problems, a similarity transformation occurs when several independent variables appear in specific combinations, in flow geometries involving infinite or semi-infinite surfaces. This leads to "rescaling", or the introduction of dimensionless groups, which converts the original PDEs into ODEs.

At least, that's as far as I understand the subject.

 

[hammadmunawar]

Similarity transformation results in a diagonal matrix. As you must know that diagonal matrices make calculations easier.

Similar matrices share a number of properties:-

They have the same rank
They have the same determinant
They have the same eigenvalues
They have the same characteristic polynomial
(and some other properties)

So, it is mostly beneficial to convert a matrix to its similar diagonal matrix, and perform calculations