## The Inverse of an Orthogonal Matrix is its Transpose.

### November 30, 2010

This is a really sweet deal. In particular, if we know that our matrix is orthogonal, we can cut down on time finding the inverse *significantly*. Combined with the spectral theorem (which states that if the matrix is symmetric, there is an *orthogonal* matrix such that is a diagonal matrix with entries the eigenvalues) this gives us a tool for finding diagnalizations of matrices.

All I want to do is state and prove this. Before we do, let’s just define the transpose really quickly.

**Definition: **The *transpose* of an matrix is the matrix denoted whose -th entry is the -th entry of .

In other words, you flip the indices. Each becomes . Now, this gives us a really nice way to compute the standard dot product.

**Claim:** For vectors of the same dimension, we have that where the right-hand-term is just matrix multiplication.

Why is this? Well, just think about it for a second and look at this example.

Then we have the dot product is just . But what is ?

and matrix multiplication gives us

same as above. This should give you a clear picture of what is going on here. Now the main theorem.

**Theorem: ** is orthogonal (all of its columns are *orthonormal, not just orthogonal*), if and only if we have .

*Proof.* This proof is not as hard as you’d expect, and uses only one main idea to make the proof elegant: the claim above. We first do a little prep work to get down to the "main matrix" and then the if and only if will follow. Let’s let be defined as follows, with each :

Now, note that is equal to almost the same thing, except the columns are now the rows, and we’ve turned them up. Check this. It’s equal to:

Now we just multiply these two together. We obtain, as you can check:

But, because of our claim above, we can simplify this nightmarish mess.

This is the "main matrix" that I was describing above. This matrix is the meat of this proof. Here’s why.

Suppose that is orthogonal. Then for each , and if . Replacing these values in the "main matrix", we get:

Which is the identity matrix. By the uniqueness of inverses for matrices, this means that .

Now suppose the converse is true: that . Well, what does this say? It says that for our "main matrix" that the diagonals are all 1 and the other elements are 0; in other words, for each and if . The latter expression tells us the columns are orthogonal, and the former tells us that these columns are actually orthonormal. Hence is an orthogonal matrix.

This is excellent. Thank you.

This is magnificent! I was readying myself for some head-banging while trying to find out why the inverse is the transpose… and then you came and saved the day! A bow of gratitude for you, dear sir!

Nice proof. Very easy to follow.

it helped a lot and saved our time

vrushali

After reading it now I understood it very conveniently. Thanks sir.

wow!this is very well written! thank u!

Vow. Thanks a Ton. This was exactly what I was looking for! :)

Sairam S

Great blog! Do you have any tips and hints for aspiring writers?

I’m planning to start my own blog soon but I’m a little lost on everything.

Would you recommend starting with a free platform like WordPress or go for a paid option?

There are so many choices out there that I’m completely

overwhelmed .. Any recommendations? Thanks!

I almost never comment on here anymore, but I dig when people want to start their own blogs!

There’s two steps here that I feel are the most important: (1) Only worry about the content; (2) Don’t sweat the small stuff.

I started on wordpress, and it was fantastic — it’s great blogging practice since it’s all free, it looks pretty, and you can really find your voice with it. It’s also a fairly nice platform to do math on, due to some of the nice LaTeX additions they have. Having said that, at some point, you may want to upgrade to something paid if you want more control over what you’re doing (for me, that took around 4 – 5 years; wordpress was fine in the meantime). But don’t worry about that now. See if you can make good content first. The rest will follow.

Don’t spend too much time worrying about how your blog looks, what your blog is called, etc. (all this can be changed later!), just start writing.

So, go. Start writing.

Thanks alot.

you don’t ‘prove’ the claim though..?

So, is it inverse or left inverse?

Reblogged this on soksereyoudomlife and commented:

TD I solve , exercise 4 solved :D

Lovely proof, thank you.