Calculating angles between matrices

[spicytaco]

Hey all,

I was hoping someone could explain to me how to calculate the angle between matrices, ie. two square matrices

[ 2 0
0 -1]

and

[0 1
1 3^(1/2)]

under the inner product <A|B> = trace (A^TB)

Also, how would you go about determining an angle between x and y when they are functions, ie. x = f(x) = x^2 +2 and y=(g(x)=x^3 -7x, under the inner product below:

⟨f |g⟩ =
1
∫ f (x)g(x)dx.
−1

I already know how to determine angle using cos theta = (x^Ty)/ ||x|| ||y|| but does this only work for column and row matrices?

 

[chogg]

Any time you have an inner product, you can use it to define angles between elements using the usual formula:
$$\cos\theta(a,b) = \frac{\langle a | b \rangle}{\sqrt{\langle a | a \rangle \langle b | b \rangle}}$$
This works equally well for your matrices and for your functions.

Whether that angle tells you anything useful for the problem you're trying to solve is another question. :)

 

[spicytaco]

Thanks so much!

 

[HallsofIvy]

For "vectors in space", two or three dimensions, you can use trigonometry to prove that the dot product of two vectors, u and v, are given by $u\cdot v= |u||v|cos(\theta)$ where |u| and |v| are the lengths of the two vectors and $\theta$ is the angle between them. For more abstract vector spaces, we define the length of a vector, v, to be $\sqrt{<v, v>}$ and the angle between u and v to be given by
$$cos(\theta)= \frac{<u, v>}{|u||v|}$$

 

[unusualname]

In spicytaco's matrix example the angle is the same as the angle for the two vectors (2,0,0,-1) and (0,1,1, 3^1/2) in R^4 with usual euclidean norm.

(2,0,0,-1).(0,1,1,3^1/2) = 0 + 0 + 0 - 3^1/2

|(2,0,0,-1)| = 5^1/2
|(0,1,1,3^1/2)| = 5^1/2

so cos(theta) = (-3^1/2)/5 (theta ~= 110 degrees)

this is why the inner product is defined as trace(A^T.B), for nxn matrices you get R^(n^2) euclidean space with two vectors defined by joining the columns of A and the other by the columns of B.

For the functions example there isn't a "natural" analogy but the answer has been given above, you just plug in the formula for the inner product into the equation for cos(theta)