Can someone explain to me, in words, what eigenvalues and eigenvectors are? It helps me to understand what they signify, so if anyone can explain them to me, I would be very grateful!

Question

turingtest · Answer

An Eigenvector can be understood as a vector that is invariant under a certain transformation (remember a vector transformation can be written as a matrix). In the case of this Wikipedia article, they show the blue horizontal "Eigenvector" for the transformation matrix that would correspond to image shearing. The red vector changes under the transformation, and so is not an Eigenvector of the shearing operation. http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

turingtest · Answer

The red vector is still likely an Eigenvector for a different matrix, though, for instance one that corresponds to spinning the Mona Lisa image would change the blue vector (so it would not be an Eigenvector for the rotation matrix) but the red vector would stay the same, and therefor be an Eigenvector of that transformation.

turingtest · Answer

as is also explained, the Eigenvalue corresponds to any magnitude change in an Eigenvector for a particular matrix. i.e. how much longer or shorter it gets under that transformation.

turingtest · Answer

That's the most physical interpretation I can give, I hope it helps!

turingtest · Answer

Basically, if you put an Eigenvector for a matrix through that matrix, its direction does not change, and its magnitude gets multiplied by the Eigenvalue.

snowfire · Answer

I don't think I understand your last comment. Are you saying that an eigenvector is actually just a matrix that when multiplied by the original matrix, gives you some multiple of the original matrix?

turingtest · Answer

technically all vectors are matrices, but maybe it helps if you replace the word "matrix" by "transformation"

Take a vector (an arrow drawn on the Mona Lisa) and transform it by multiplying it with that particular matrix for which it is an Eigenvector. It will not change direction, other than perhaps become negative, and will change in size by a factor of the Eigenvalue.

snowfire · Answer

So a transformation can be thought of as an eigenvector? Where the eigenvalue scales it up or down along that vector's direction?

turingtest · Answer

The transformation is the matrix. The eigenvector is the vector that won't change direction when you multiply it by that matrix.

snowfire · Answer

So to that eigenvector, the transformation is basically just a scalar?

turingtest · Answer

exactly :)

snowfire · Answer

And the eigenvalue is how big of a value that scalar represents?

turingtest · Answer

precisely

snowfire · Answer

Oh it makes so much more sense now! I could have spent all day looking at wikipedia and it wouldn't have done anything for me. Thank you so much, your effort towards helping me is greatly appreciated.

turingtest · Answer

Happy to help. Really only the drawing of the Mona Lisa is helpful for this discussion, hence the link.
Welcome!

snowfire · Answer

Yeah I can see its purpose now, thanks again