Proving diagonal entries of R in QR decomposition nonzero

[derryck1234]

Homework Statement

Prove that the diagonal entries of R in the QR decomposition are non-zero.

Homework Equations

ui = qi + <ui, q1>q1 + <ui, q2>q2 + ... + <ui, qi-1>qi-1 => From the general equation for the Gram-Schmidt process.

<ui, qi> => From the diagonal entries of QR.

The Attempt at a Solution

I have the solution from another website. I just want help regarding the part where the properties of the inner product are used in the following step:

<qi + <ui, q1>q1 + <ui, q2>q2 ... + <ui, qi-1>qi-1, qi>

to

<qi, qi> + <ui, q1><q1, qi> + ... + <ui, qi-1><qi-1, qi>

I have not seen a property of the inner product specifying that:

<<u, v>v,w> = <u,v><v,w>

Or, have I written it correctly?

Please help.

Thanks

 

[mighty2000]

Hello,

You are correct in saying that the property you mentioned is not a direct property of the inner product. However, it can be derived from the properties of the inner product and the properties of scalar multiplication. Here is the proof for your reference:

Let u, v, and w be vectors and c be a scalar. Then, we have:

<u + cv, w> = <u, w> + c<v, w> [Property of the inner product]

Now, let u = v in the above equation. Then, we get:

<2v, w> = <v, w> + 2<v, w> [Substituting u = v]

<2v, w> = 3<v, w> [Simplifying]

<2v, w> = 3<v, w> [Distributing the scalar multiplication]

2<v, w> = 3<v, w> [Dividing both sides by 2]

<v, w> = 0 [Subtracting 2<v, w> from both sides]

Therefore, we can see that if u = v, then <u, w> = 0, which is the property you mentioned. This can be extended to any scalar multiple of v, including <ui, qi-1>, where i = 1, 2, ..., n.

I hope this helps! Let me know if you have any further questions.