18.06 Linear Algebra, Spring 2010
Transcript – Lecture 17
OK, here's the last lecture in the chapter on orthogonality.
So we met orthogonal vectors, two vectors, we met orthogonal subspaces, like the
row space and null space. Now today we meet an orthogonal basis, and an
orthogonal matrix. So we really -- this chapter cleans up orthogonality.
And really I want -- I should use the word orthonormal.
Orthogonal is -- so my vectors are q1,q2 up to qn -- I use the letter "q", here, to
remind me, I'm talking about orthogonal things, not just any vectors, but orthogonal
ones. So what does that mean? That means that every q is orthogonal to every other
q.
It's a natural idea, to have a basis that's headed off at ninety-degree angles, the
inner products are all zero. Of course if q is -- certainly qi is not orthogonal to itself.
But there we'll make the best choice again, make it a unit vector.
Then qi transpose qi is one, for a unit vector.
The length squared is one. And that's what I would use the word normal. So for this
part, normalized, unit length for this part.
OK. So first part of the lecture is how does having an orthonormal basis make things
nice? It certainly does. It makes all the calculations better, a whole lot of numerical
linear algebra is built around working with orthonormal vectors, because they never
get out of hand, they never overflow or underflow. And I'll put them into a matrix Q,
and then the second part of the lecture will be suppose my basis, my columns of A
are not orthonormal.
How do I make them so? And the two names associated with that simple idea are
Graham and Schmidt. So the first part is we've got a basis like this.
Let's put those into the columns of a matrix.
So a matrix Q that has -- I'll put these orthonormal vectors, q1 will be the first
column, qn will be the n-th column.
And I want to say, I want to write this property, qi transpose qj being zero, I want to
put that in a matrix form. And just the right thing is to look at Q transpose Q. So this
chapter has been looking at A transpose A. So it's natural to look at Q transpose Q.
And the beauty is it comes out perfectly. Because Q transpose has these vectors in
its rows, the first row is q1 transpose, the nth row is qn transpose. So that's Q
transpose.
And now I want to multiply by Q.