Linear Algebra; Explain why [tex]M^2=(\vec{v} \cdot \vec{w})M[/tex]

[ksdhart2]

Hi all,

I'm having difficulty with a problem in my Linear Algebra class. I've been working on it and am struggling to see what the next step ought to be. The full text of the problem is:

3a) Let \(\displaystyle \vec{v}\) and \(\displaystyle \vec{w}\) be two n x 1 matrices ("column vectors"). Define \(\displaystyle M=\vec{v}\vec{w}^T\), where \(\displaystyle \vec{w}^T\) is the transpose of \(\displaystyle \vec{w}\). Explain why \(\displaystyle M^2=\left( \vec{v} \cdot \vec{w} \right) M\).

The first thing I did was rewrite the two equation for M², like so:

\(\displaystyle M^2=MM=\vec{v}\vec{w}^T \vec{v}\vec{w}^T\)

I know that \(\displaystyle \vec{v} \cdot \vec{w}=\vec{v}^T\vec{w}\). Making the appropriate substitution gives:

\(\displaystyle \left( \vec{v} \cdot \vec{w}\right)M=\left( \vec{v}^T\vec{w} \right) \left( \vec{v}\vec{w}^T \right)\)

And I know something about this is wrong, because that multiplication doesn't actually exist, due to a mismatch between the number of rows and columns. But I know the overall product does exist because the dot product is a scalar value. I just don't know where I went wrong.

For completeness' sake, I'll include the two hints the professor gave us at the end of this homework packet:

For problem 3a:
Hint #1) A dot product can be expressed as a matrix multiplication. \(\displaystyle \vec{v} \cdot \vec{w}=\vec{v}^T\vec{w}\) or since \(\displaystyle \vec{v} \cdot \vec{w}=\vec{w} \cdot \vec{v}\), as \(\displaystyle \vec{w}^T\vec{v}\)

Hint #2) Use the associate law of multiplication, along with scaling properties (i.e. \(\displaystyle A(\lambda B=\lambda AB\) where \(\displaystyle \lambda\) is a scalar)

Given the first hint, I strongly believe my work is at least in the general area of the solution, but I'm just missing that one little thing that makes it all click in my brain. Any help would be greatly appreciated.

 

[Ishuda]

Hi all,

I'm having difficulty with a problem in my Linear Algebra class. I've been working on it and am struggling to see what the next step ought to be. The full text of the problem is:



The first thing I did was rewrite the two equation for M², like so:

\(\displaystyle M^2=MM=\vec{v}\vec{w}^T \vec{v}\vec{w}^T\)

I know that \(\displaystyle \vec{v} \cdot \vec{w}=\vec{v}^T\vec{w}\). Making the appropriate substitution gives:

\(\displaystyle \left( \vec{v} \cdot \vec{w}\right)M=\left( \vec{v}^T\vec{w} \right) \left( \vec{v}\vec{w}^T \right)\)

And I know something about this is wrong, because that multiplication doesn't actually exist, due to a mismatch between the number of rows and columns. But I know the overall product does exist because the dot product is a scalar value. I just don't know where I went wrong.

For completeness' sake, I'll include the two hints the professor gave us at the end of this homework packet:



Given the first hint, I strongly believe my work is at least in the general area of the solution, but I'm just missing that one little thing that makes it all click in my brain. Any help would be greatly appreciated.

Let
\(\displaystyle \lambda=\vec{w}^T \vec{v}=\vec{v}^T \vec{w}\)
and write
\(\displaystyle M^2=MM=\vec{v}\vec{w}^T \vec{v}\vec{w}^T=\vec{v}\, (\vec{w}^T\, \vec{v})\, \vec{w}^T\)

Does that help?

 

[ksdhart2]

Let
\(\displaystyle \lambda=\vec{w}^T \vec{v}=\vec{v}^T \vec{w}\)
and write
\(\displaystyle M^2=MM=\vec{v}\vec{w}^T \vec{v}\vec{w}^T=\vec{v}\, (\vec{w}^T\, \vec{v})\, \vec{w}^T\)

Does that help?

Yes, that solved my woes. Thank you! I knew it had to be just one small detail, like changing where the parentheses go, that would make it clear. Picking up from where you left off:

\(\displaystyle \vec{v}\, (\vec{w}^T\, \vec{v})\, \vec{w}^T=\vec{v} \left(\vec{v} \cdot \vec{w} \right)\vec{w}^T\)

By the associative property, I can move the scalar dot product out front:

\(\displaystyle \vec{v} \left(\vec{v} \cdot \vec{w} \right)\vec{w}^T=\left(\vec{v} \cdot \vec{w}\right) \vec{v} \vec{w}^T=\left(\vec{v} \cdot \vec{w}\right)M \therefore M^2=\left( \vec{v} \cdot \vec{w}\right)M\)