Solving quadratic matrix equation for X X2=[1a01] where a∈R0. Solve for matrix X.

Question

Solving quadratic matrix equation for X  X2=[1a01]  where a∈R0. Solve for matrix X.

Kyler Jacobson · Accepted Answer

Step 1LetM(a)=[1a01].Note M(a) has one eigenvalue: 1, and is not diagonalisable. This means that, if X2=M(a), then X has eigenvalue 1 or -1, but not both (if it had both, then X would be diagonalisable, and so would X2=M(a))We also note that any eigenvectors of X will be an eigenvector for X2, and X2 has only the eigenvector[10]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[10]on the right reveals the first column of a 2×2 matrix. Therefore, the first column of X isX[10]=±1⋅[10]=[±10],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:[±1∗0±1]=±[1∗01],where the three ±′s are all the same. That is,X=±M(b)for some b. Note that (±M(b))2=M(b)2, so we are looking for b such that[1b01]2=[1a01],which was solved by Angel in their answer: b=a2. This gives us precisely two solutions:X=[1a201],−[1a201].

becky4208fj · Accepted Answer

Step 1  Let X=(I+N) 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,  X2=I+N+N+N2=I+2N  the answer is trivially obtained from that.

Solving quadratic matrix equation for X \(X^2 = \begin{bmatrix}1&a\\0&1\\\end{bmatrix}\) where a\in\frac{\mathbb{R}}{0}. Solve

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