18.06 Linear Algebra, Spring 2010
Transcript – Lecture 32
OK, here we go with, quiz review for the third quiz that's coming on Friday. So, one
key point is that the quiz covers through chapter six. Chapter seven on linear
transformations will appear on the final exam, but not on the quiz. So I won't review
linear transformations today, but they'll come into the full course review on the very
last lecture.
So today, I'm reviewing chapter six, and I'm going to take some old exams, and I'm
always ready to answer questions.
And I thought, kind of help our memories if I write down the main topics in chapter
six.
So, already, on the previous quiz, we knew how to find eigenvalues and
eigenvectors. Well, we knew how to find them by that determinant of A minus
lambda I equals zero. But, of course, there could be shortcuts. There could be, like,
useful information about the eigenvalues that we can speed things up with. OK.
Then, the new stuff starts out with a differential equation, so I'll do a problem. I'll do
a differential equation problem first. What's special about symmetric matrices? Can
we just say that in words? I'd better write it down, though.
What's special about symmetric matrices? Their eigenvalues are real. The
eigenvalues of a symmetric matrix always come out real, and there always are
enough eigenvectors. Even if there are repeated eigenvalues, there are enough
eigenvectors, and we can choose those eigenvectors to be orthogonal.
So if A equals A transposed, the big fact will be that we can diagonalize it, and those
eigenvector matrix, with the eigenvectors in the column, can be an orthogonal
matrix. So we get a Q lambda Q transpose. That, in three symbols, expresses a
wonderful fact, a fundamental fact for symmetric matrices. OK.
Then, we went beyond that fact to ask about positive definite matrices, when the
eigenvalues were positive.
I'll do an example of that. Now we've left symmetry.
Similar matrices are any square matrices, but two matrices are similar if they're
related that way.
And what's the key point about similar matrices? Somehow, those matrices are
representing the same thing in different basis, in chapter seven language.
In chapter six language, what's up with these similar matrices? What's the key fact,
the key positive fact about similar matrices? They have the same eigenvalues. Same
eigenvalues.