Question

How do you guys visualize eigenspaces, and how they relate to their matrices? For example, diagonalizing matrices involves multiplying a diagonal matrix of eigenvalues by its orthonormalized eigenspaces (eigenvectors) and their transpose, A = Q*D*Q^T. Does anyone of a good image of this, or other eigenstuff, intuitively in their minds they care to share? Would appreciate :) Linear Algebra is my last final on Wednesday; I can taste the freedom @_@

Answer 1

My knowledge of Linear Algebra is pretty elementary as well... So as far as I know, the act of diagonalizing matrices can be thought as changing basis of your vector space. Eigenspaces are a special subspace of your vector space because, when you apply A to the the Eigenspace, everything remains in the Eigenspace, while if you apply A to any other subspace which isn't an Eigenspace, then the image is not the exact same subspace as the subspace. Note that the Eigenspace still remains the same no matter the change of basis (i.e. it is basis invariant)

Answer 2

The easiest Eigenspace I think of is when the linear transformation is a shear

Answer 3

Whoa, shear transformations seem to still be outside my realm but your other explanation is pretty useful :P

Answer 4

Shears are something i learnt in secondary/high school but I guess it may not be very useful if you haven't come across it yet! Pretty simple stuff really: http://mathworld.wolfram.com/Shear.html

Answer 5

Hmmm, if the vector space is \(\mathbb{R}^3\), then if you have a rotation matrix, the axis of rotation would be an eigenspace.