18.06 Linear Algebra, Spring 2010
Transcript – Lecture 27
OK, this is the lecture on positive definite matrices.
I made a start on those briefly in a previous lecture.
One point I wanted to make was the way that this topic brings the whole course
together, pivots, determinants, eigenvalues, and something new- four plot instability
and then something new in this expression, x transpose Ax, actually that's the guy to
watch in this lecture.
So, so the topic is positive definite matrix, and what's my goal? First, first goal is,
how can I tell if a matrix is positive definite? So I would like to have tests to see if
you give me a, a five by five matrix, how do I tell if it's positive definite? More
important is, what does it mean? Why are we so interested in this property of
positive definiteness? And then, at the end comes some geometry. Ellipses are
connected with positive definite things. Hyperbolas are not connected with positive
definite things, so we- it's this, we, there's a geometry too, but mostly it's linear
algebra and -- this application of how do you recognize 'em when you have a minim
is pretty neat. OK. I'm gonna begin with two by two.
All matrices are symmetric, right? That's understood; the matrix is symmetric, now
my question is, is it positive definite? Now, here are some -- each one of these is a
complete test for positive definiteness. If I know the eigenvalues, my test is are they
positive? Are they all positive? If I know these -- so, A is really -- I look at that
number A, here, as the, as the one by one determinant, and here's the two by two
determinant.
So this is the determinant test.
This is the eigenvalue test, this is the determinant test. Are the determinants
growing in s- of all, of all end, sort of, can I call them leading submatrices, they're
the first ones the northwest, Seattle submatrices coming down from from there, they
all, all those determinants have to be positive, and then another test is the pivots.
The pivots of a two by two matrix are the number A for sure, and, since the product
is the determinant, the second pivot must be the determinant divided by A.
And then in here is gonna come my favorite and my new idea, the, the, the the one
to catch, about x transpose Ax being positive. But we'll have to look at this guy. This
gets, like a star, because for most, presentations, the definition of positive
definiteness would be this number four and these numbers one two three would be
test four. OK.
Maybe I'll tuck this, where, you know, OK. So I'll have to look at this x transpose Ax.
Can you, can we just be sure, how do we know that the eigenvalue test and the