18.06 Linear Algebra, Spring 2010
Transcript – Lecture 25
-- one and -- the lecture on symmetric matrixes.
So that's the most important class of matrixes, symmetric matrixes. A equals A
transpose.
So the first points, the main points of the lecture I'll tell you right away. What's
special about the eigenvalues? What's special about the eigenvectors? This is -- the
way we now look at a matrix. We want to know about its eigenvalues and
eigenvectors and if we have a special type of matrix, that should tell us something
about eigenvalues and eigenvectors. Like Markov matrixes, they have an eigenvalue
equal one.
Now symmetric matrixes, can I just tell you right off what the main facts -- the two
main facts are? The eigenvalues of a symmetric matrix, real -- this is a real
symmetric matrix, we -- talking mostly about real matrixes. The eigenvalues are also
real.
So our examples of rotation matrixes, where -- where we got E- eigenvalues that
were complex, that won't happen now.
For symmetric matrixes, the eigenvalues are real and the eigenvectors are also very
special. The eigenvectors are perpendicular, orthogonal, so which do you prefer? I'll
say perpendicular. Perp- well, they're both long words. Okay, right.
So -- I have a -- you should say "why?" and I'll at least answer why for case one,
maybe case two, the checking the Eigen -- that the eigenvectors are perpendicular,
I'll leave to, the -- to the book. But let's just realize what -- well, first I have to say,
it -- it could happen, like for the identity matrix -- there's a symmetric matrix.
Its eigenvalues are certainly all real, they're all one for the identity matrix. What
about the eigenvectors? Well, for the identity, every vector is an eigenvector.
So how can I say they're perpendicular? What I really mean is the -- they -- this
word are should really be written can be chosen perpendicular.
That is, if we have -- it's the usual case.
If the eigenvalues are all different, then each eigenvalue has one line of eigenvectors
and those lines are perpendicular here. But if an eigenvalue's repeated, then there's
a whole plane of eigenvectors and all I'm saying is that in that plain, we can choose
perpendicular ones. So that's why it's a can be chosen part, is -- this is in the case of
a repeated eigenvalue where there's some real, substantial freedom.
But the typical case is different eigenvalues, all real, one dimensional eigenvector
space, Eigen spaces, and all perpendicular.