MHB Antisymmetry Invariant Under Similarity Orthogonal Transforms

[ognik]

Hi - the text is very brief on similarity transforms and wiki etc. a bit beyond where I am. In fact I think I am muddling a few things up, so I have a few questions around this topic please:
1) I'd appreciate a 'beginners' explanation of similarity transforms, what they really are and what they are most useful for?
2) Are all similarity transforms not orthogonal?
3) I think that an antisymetric matrix cannot be orthogonal, but I get the impression they are related by more than the '-' sign?
4) Finally an exercise where I'd appreciate a starter tip - if not already included in the answers above:"Show that the property of antisymetry is invariant under orthogonal similarity transformations" . All help much appreciated :-)

 

[I like Serena]

Hi - the text is very brief on similarity transforms and wiki etc. a bit beyond where I am. In fact I think I am muddling a few things up, so I have a few questions around this topic please:
1) I'd appreciate a 'beginners' explanation of similarity transforms, what they really are and what they are most useful for?
2) Are all similarity transforms not orthogonal?
3) I think that an antisymetric matrix cannot be orthogonal, but I get the impression they are related by more than the '-' sign?
4) Finally an exercise where I'd appreciate a starter tip - if not already included in the answers above:"Show that the property of antisymetry is invariant under orthogonal similarity transformations" . All help much appreciated :-)

Hey ognik!

1) A similarity transformation of a matrix A, is where you first apply some matrix to A and afterwards "undo" the effect of that matrix by applying its inverse.
A different name for it is conjugation.

It's one of the basic principles to solve many puzzles.
Take for instance Rubik's cube.
A conjugate move is one of the form $XYX^{-1}$. It is also called a "setup move".
That is, $X^{-1}$ is the setup for a move $Y$. Afterwards the setup move is reversed, leaving only the effect of $Y$ that has been tweaked a bit.

Two matrices are called similar if there is a similarity transformation that transforms the one into the other.
And here's where to power of similarity manifests: most properties of those similar matrices are identical.
For instance, they have the same determinant, the same eigenvalues, the same trace, and so on.2) A similarity transform is typically not orthogonal - it's not even a matrix transform. It's a transformation that consists of 2 matrices: one that is applied before, and its inverse that is applied after.

Of course a similarity transform can be built from an orthogonal matrix.3) Pick $$(^{0\ -1}_{1\ \phantom{-}0})$$.
Is it antisymmetric? Is it orthogonal?4) Suppose A is antisymmetric and B is orthogonal. Then $BAB^{-1}$ is such a similarity transformation.
In index notation it is:
$$(bab^{-1})_{ij} = \sum_{k,l} b_{ik}a_{kl}b^{-1}_{lj}$$
Can you prove that it is equal to:
$$-(bab^{-1})_{ji}$$
? (Wondering)

 

[ognik]

Great, thanks!
1) Very clear. Do you perhaps have a link to an example where a matrix has been tweaked (usefully), so I can sit and contemplate it a bit?
2) Thanks, clear.
3) OK, I had read the problem as A being both antisymetric AND Orthogonal, that 'de-confuses' both 3) and 4) thanks.
Regards

 

[I like Serena]

Good! ;)

Check out:
Jordan normal form - Wikipedia, the free encyclopedia

It shows how to find a similar matrix in a normalized form so you can immediately read off all the important matrix properties.