# Inner product vs. dot product

1. May 18, 2006

### dimensionless

Is there any difference between an inner product and a dot product?

2. May 18, 2006

### Dragonfall

3. May 18, 2006

### arildno

Yes:
The dot product is an inner product, whereas "inner product" is the more general term.

EDIT:
I'm getting old. But then again, how could I ever compete with young, strong dragons swooping down on its prey?

4. May 18, 2006

### dimensionless

What would be an example of an inner product that it not a dot product?

5. May 18, 2006

### arildno

Well, as I'm used to it the term "dot product" is usually reserved for an operation on a finite Euclidean space; I haven't seen the term "dot product" being used for inner products defined on function spaces, for example.

6. May 18, 2006

### LeonhardEuler

Just expanding on what arildno said, one specific example is the Frobenius inner product of two matrices. It is defined by:
<A,B>=tr(B*A)
Where B* is the conjugate transpose, or adjoint of B.

7. May 18, 2006

### TD

For continous functions from [a,b] to R, you can define an inner product <.,.> as:

$$\left\langle {f,g} \right\rangle = \int\limits_a^b {f\left( x \right)g\left( x \right)dx}$$

This is also an example on that Mathworld page.

8. May 18, 2006

### Hurkyl

Staff Emeritus
(a, b) . (c, d) = ac + 2bd
(a, b) . (c, d) = (a+b)(c+d) + (a-b)(c-d)

(a) . (b) = 2ab

9. May 19, 2006

### dimensionless

Are these postulates?

10. May 19, 2006

### arildno

No, you can show that they satisfy the PROPERTIES of the inner product:
To take the first:
1. (a,b).(a,b)=a^2+2b^2>0 unless a=b=0
2. (a,b).(c,d)=ac+2bd=ca+2db=(c,d).(a,b)
and so on with the rest of an inner product's properties.

11. May 19, 2006

### Staff: Mentor

I wish someone had explained that difference to me 30+ years ago, when I was first learning about vectors and multidimensional analysis. I got to university thinking that 'dot' and 'inner' product were the same, and it was terribly confusing when an inner product was introduced with a different meaning than a dot product.

12. May 19, 2006

### arildno

The way I sort of organized the concepts to myself was like this:
General terms:
Inner/Outer products
Special terms:
Dot/cross products
Scalar/vector products

This was at least helpful for me.

13. May 19, 2006

### Dragonfall

There's an outer product now?

14. May 19, 2006

### daveb

Yup. It's commonly called the tensor or direct product.

15. May 19, 2006

### Hurkyl

Staff Emeritus
No -- they are definitions of three different inner products. The first two are on R², and the third is on R.

16. May 19, 2006

### HallsofIvy

Staff Emeritus
The theoretical "meat" of the Gram-Schmidt orthogonaliztion process is that any inner product is a dot product in some basis. Given an inner product, choose a basis and use Gram-Schmidt to derive an orthonormal basis {e1, e2,...,en}. For any vectors u,v, write u= a1e1+ ... , v= b1e1+... . Then the inner product <u,v>= a1b1+... + anbn.

Take a look at Hurkyl's examples:
(a, b) . (c, d) = ac + 2bd
(a, b) . (c, d) = (a+b)(c+d) + (a-b)(c-d)

(a) . (b) = 2ab
Starting from the basis {(1, 0), (0,1)}, (or just (1) for the third example) use Gram-Schmidt with each of these inner products to derive an orthonormal basis and show that the inner product is the dot product in that basis.