Prove that if A and B are n × n orthogonal matrices, then A...

Transcript

00:01 So if a and b orthogonal matrices, what that means is the following.

00:08 So that means if we were to take it and multiply it by its transpose, that would be the same thing as multiplying by its transpose, and that should go ahead and give us the identity matrix.

00:20 So this is what we're starting with.

00:23 So essentially what we want to show, we want to show that ab, the transpose of that times ab.

00:32 Is equal to ab times the transpose of ab is identity matrix, as well as where it's flipped around.

00:47 Let's go ahead and do this here.

00:53 So first proof.

00:56 Actually, let me write in blue.

00:58 So proof assume a and b are elements of r in by n, which is just saying they're in by n matrix.

01:12 And orthogonal.

01:19 So now let's look at what a -b times a -b -t is.

01:28 Well, first, if we have two things being multiplied together and we take the transpose, what we can do is rewrite this as b transpose times a transpose, like that.

01:46 And now we can go ahead and group these to be where it is a times b transpose times a transpose.

01:58 And i'm going to put a little asterisk here, just why i don't have to keep restating this over and over again.

02:06 But now this, b times b transpose, should be just the identity matrix, since we're assuming b is orthogonal.

02:15 So this is going to be a times.

02:19 Here, let me do this in green.

02:22 So the identity matrix times a transpose, and this is by the fact that it's orthogonal.

02:31 And then anything times identity matrix should just be itself...

Question

Prove that if $A$ and $B$ are $n \times n$ orthogonal matrices, then $A B$ and $B A$ are orthogonal.

Question

Prove that if $A$ and $B$ are $n \times n$ orthogonal matrices, then $A B$ and $B A$ are orthogonal.

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