A^2 = A, is λ (eigenvalue) in {0,1}

[Enemy of my enemy]

Hello,
I have question about this problem. I tried to find matrix (if A² = A) to find out if eigenvalue can be between 0 and 1... but I only find 1.
So can I find any other for this conditions?

 

[HallsofIvy]

Your title says "in {0, 1}" but in your post you say "can be between 0 and 1". Those are NOT the same thing! "Between 0 and 1" would be (0, 1). {0, 1} means that it must be either 0 or 1.

An important property of eigenvalues is that "Every matrix satisfies it own eigenvalue equation (or "characteristic equation")". Since \(\displaystyle A^2= A\), if \(\displaystyle \lambda\) is an eigenvalue, then \(\displaystyle \lambda^2= \lambda\), \(\displaystyle \lambda^2- \lambda= \lambda(\lambda- 1)= 0\) so \(\displaystyle \lambda= 0\) or \(\displaystyle \lambda= 1\). The eigenvalues are 0 and 1, not "between 0 and 1".

 

[Enemy of my enemy]

Thank you sir,
I did very stupid mistake. You are right!