Solving quadratic matrix equation for X X2=[1a01] where a∈R0. Solve for matrix X.
gea3stwg
Answered
2022-01-30
Solving quadratic matrix equation for X
where . Solve for matrix X.
Answer & Explanation
Kyler Jacobson
Expert
2022-01-31Added 8 answers
Step 1 Let 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:
becky4208fj
Expert
2022-02-01Added 10 answers
Step 1
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.