Let \[ A=\begin{pmatrix} 2&-1&0\\ -1&2&-1\\ 0&-1&2 \end{pmatrix}. \] Show that every real matrix \(B\) satisfying \(AB=BA\) has the form \[ B=aI+bA+cA^2 \] for some \(a,b,c\in\mathbb R\).
Solution
1. The characteristic polynomial of \(A\) has the three distinct roots\[ \lambda_1=2-\sqrt2, \qquad \lambda_2=2, \qquad \lambda_3=2+\sqrt2. \]
Set \[ t=2-\lambda. \] Then \[ \det(A-\lambda I) = \det \begin{pmatrix} t&-1&0\\ -1&t&-1\\ 0&-1&t \end{pmatrix}. \] Expanding along the first row, \[ \begin{aligned} \det(A-\lambda I) &= t(t^2-1)-t\\ &= t(t^2-2). \end{aligned} \] Thus \[ t=0,\ \pm\sqrt2, \] which gives the three displayed values of \(\lambda\).
2. The space \(\RR^3\) is the direct sum of the three one-dimensionaleigenspaces\[ E_{\lambda_1}\oplus E_{\lambda_2}\oplus E_{\lambda_3}. \]
Step 1 gives three distinct eigenvalues of a \(3\times3\) matrix. Their eigenvectors are linearly independent, so the three eigenspaces are each one-dimensional and their direct sum is all of \(\RR^3\).
3. If \(AB=BA\), then every eigenspace \(E_{\lambda_j}\) is\(B\)-invariant.
If \[ Av=\lambda_jv, \] then \[ A(Bv) = B(Av) = \lambda_jBv. \] Hence \[ Bv\in E_{\lambda_j}. \]
4. There are real numbers \(\beta_1,\beta_2,\beta_3\) such that\[ Bv=\beta_jv \]for every\[ v\in E_{\lambda_j}. \]
By step 2, each \(E_{\lambda_j}\) is one-dimensional. Step 3 shows that \(B\) restricts to an endomorphism of that line, so it acts there by a scalar. Since both the line and \(B\) are real, that scalar is real.
5. There is a real polynomial\[ p(t)=a+bt+ct^2 \]such that\[ p(\lambda_j)=\beta_j \qquad j=1,2,3. \]
1. Lagrange interpolation therefore gives a unique polynomial of degreeat most \(2\) taking the prescribed real values \(\beta_j\) at those threepoints. Its coefficients are real because all interpolation data are real.
6. One has\[ B=p(A). \]
Let \[ v\in E_{\lambda_j}. \] Then step 4 gives \[ Bv=\beta_jv, \] while step 5 gives \[ p(A)v = p(\lambda_j)v = \beta_jv. \] Thus \(B\) and \(p(A)\) agree on every eigenspace. By step 2, those eigenspaces span \(\RR^3\), so the two linear maps are equal.
7. Therefore\[ \boxed{B=aI+bA+cA^2} \]for some real numbers \(a,b,c\).
Expand \(p(A)\) using the polynomial from step 5 and apply step 6.
8. Q.E.D.
Step 7 is the required form.