Solving quadratic matrix equation for X X2=[1a01] where a∈R0. Solve for matrix X.
Solving quadratic matrix equation for X
where . Solve for matrix X.
Answer & Explanation
Note M(a) has one eigenvalue: 1, and is not diagonalisable. This means that, if , then X has eigenvalue 1 or -1, but not both (if it had both, then X would be diagonalisable, and so would )
We also note that any eigenvectors of X will be an eigenvector for , and has only the eigenvector
Thus, X must have only this eigenvector as well. (Of course, when I say a matrix has only one eigenvector, I mean that there is one eigenspace, and it is one-dimensional, so we can only find one linearly independent eigenvector.)
Now, multiplication by
on the right reveals the first column of a matrix. Therefore, the first column of X is
depending on whether the unique eigenvalue is 1 or -1. This makes the matrix X upper-triangular (due to the 0 in the bottom left corner), so the eigenvalues of X lie on the diagonal. That is, X takes the form:
where the three are all the same. That is,
for some b. Note that , so we are looking for b such that
which was solved by Angel in their answer: . This gives us precisely two solutions:
Let where I is the identy matrix and N is the nilpotent matrix with all the diagonal entries and the lower left coner being zero. Take the second power of X,
the answer is trivially obtained from that.