Difficult Problem with Matrices

[Physics lover]

Homework Statement

Let M be a colum vector (not a null vector).Let A=(M M^T)/(M^T M)
Then A is
a)involuntary b)nilvoluntary c)identity d)idempotent

Relevant Equations

Idempotent Matrix:A^2=A
Involuntary Matrix:A^2=I
Nilvoluntary Matrix:A^2=0

I assumed a column vector of degree 3 and then calculated A from the given condition.But after solving it i tried to find A² and then I got stuck as none of the options seem to match.
Please help.
I think i will have to learn LATEX.

 

[archaic]

I think i will have to learn LATEX.

https://www.physicsforums.com/help/latexhelp/

 

[PeroK]

Homework Statement:: Let M be a colum vector (not a null vector).Let A=(M M^T)/(M^T M)
Then A is
a)involuntary b)nilvoluntary c)identity d)idempotent
Relevant Equations:: Idempotent Matrix:A^2=A
Involuntary Matrix:A^2=I
Nilvoluntary Matrix:A^2=0

I assumed a column vector of degree 3 and then calculated A from the given condition.But after solving it i tried to find A² and then I got stuck as none of the options seem to match.
Please help.
I think i will have to learn LATEX.

Better to start with vector of degree 2.

 

[Physics lover]

Better to start with vector of degree 2.

Ok so I did that but I am still stuck.

 

[PeroK]

ok so I did that but i am still stuck
https://www.physicsforums.com/attachments/260171

That link leads nowhere. It can't be hard to work it out for a vector $(a, b)$, surely?

 

[PeroK]

That's good so far. Now all you need is $A^2$.

 

[Physics lover]

That link leads nowhere. It can't be hard to work it out for a vector $(a, b)$, surely?

That's why i deleted it.
I didn't get what you said.Shall i take numbers in matrix instead of variabled?

 

[Physics lover]

That's good so far. Now all you need is $A^2$.

I calculated it but that's not matching with any option.

 

[Physics lover]

That's good so far. Now all you need is $A^2$.

But how will i find A²?
Shall i find out the squares of numerator and denominator separately?

 

[PeroK]

But how will i find A²?
Shall i find out the squares of numerator and denominator separately?

The denominator is just a number. You can leave that outside the matrix. Just square the matrix in the numerator and see what you get.

 

[Physics lover]

The denominator is just a number. You can leave that outside the matrix. Just square the matrix in the numerator and see what you get.

oh yes I got it.By taking (a²+b²) common I got A²=A.Thanks a lot for the help.And is it correct now?

 

[PeroK]

oh yes I got it.By taking (a²+b²) common I got A²=A.Thanks a lot for the help.And is it correct now?

That proves it for 2x2 matrices.

 

[Physics lover]

That proves it for 2x2 matrices.

so shall i try for degree 3 too?

 

[PeroK]

so shall i try for degree 3 too?

It's up to you. It's your question!

There might be a quick way to prove that for any vector dimension.

 

[Physics lover]

It's up to you. It's your question!

There might be a quick way to prove that for any vector dimension.

ok I proved it for degree 3 too.But can you tell me the general solution.
That would be a great help.

 

[PeroK]

ok I proved it for degree 3 too.But can you tell me the general solution.
That would be a great help.

I haven't looked for a general solution. It's too much effort!

 

[Physics lover]

I haven't looked for a general solution. It's too much effort!

ok no problem.
Thanks a lot for the help.