Which of the following matrices are similar over \(\RR\)? \[ A=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}, \qquad B=\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}, \qquad C=\begin{pmatrix}1&0&0\\1&1&0\\0&0&1\end{pmatrix}, \] \[ D=\begin{pmatrix}1&0&0\\1&1&0\\0&1&1\end{pmatrix}, \qquad E=\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}, \qquad F=\begin{pmatrix}0&1&1\\0&1&0\\1&0&0\end{pmatrix}. \]
Solution
1. The four matrices \(A,C,D,E\) have characteristic polynomial\[ (t-1)^3, \]whereas \(B\) and \(F\) have characteristic polynomial\[ (t-1)^2(t+1). \]
The matrices \(A,C,D,E\) are triangular with every diagonal entry equal to \(1\), so their characteristic polynomial is \((t-1)^3\).
For \(B\), \[ \det(tI-B) = \det \begin{pmatrix} t&0&-1\\ 0&t-1&0\\ -1&0&t \end{pmatrix} = (t-1)(t^2-1) = (t-1)^2(t+1). \] Similarly, \[ \det(tI-F) = \det \begin{pmatrix} t&-1&-1\\ 0&t-1&0\\ -1&0&t \end{pmatrix} = (t-1)(t^2-1). \]
2. The matrices \(A,C,D,E\) have the following Jordan data at the soleeigenvalue \(1\):\[ \begin{array}{c|c|c} \text{matrix}&\rank(M-I)&\text{nilpotency index of }M-I\\ \hline A&0&1\\ C&1&2\\ D&2&3\\ E&1&2 \end{array} \]where the zero matrix is assigned nilpotency index \(1\) for this table.
Since \(A=I\), one has \(A-I=0\).
For \(C\) and \(E\), respectively, \[ C-I= \begin{pmatrix} 0&0&0\\ 1&0&0\\ 0&0&0 \end{pmatrix}, \qquad E-I= \begin{pmatrix} 0&1&0\\ 0&0&0\\ 0&0&0 \end{pmatrix}. \] Each has rank \(1\), is nonzero, and has square zero.
Finally, \[ D-I= \begin{pmatrix} 0&0&0\\ 1&0&0\\ 0&1&0 \end{pmatrix}, \] which has rank \(2\), has nonzero square \[ (D-I)^2= \begin{pmatrix} 0&0&0\\ 0&0&0\\ 1&0&0 \end{pmatrix}, \] and has cube zero.
3. The matrices \(C\) and \(E\) are similar over \(\RR\).
By step 2, both \(C-I\) and \(E-I\) are nonzero nilpotent \(3\times3\) matrices with square zero and rank \(1\). Hence each has exactly one Jordan block of size \(2\) and one block of size \(1\) for eigenvalue \(0\). Therefore both \(C\) and \(E\) have Jordan form \[ J_2(1)\oplus[1], \] whose entries are real. Thus \(C\) and \(E\) are similar over \(\RR\).
4. No two distinct matrices among \(A,C,D,E\), except \(C\) and \(E\), aresimilar.
Similarity preserves the ranks of polynomial expressions in a matrix and the nilpotency index of \(M-I\). Step 2 distinguishes \(A\), the common class of \(C,E\), and \(D\) by these invariants.
5. The matrices \(B\) and \(F\) are not similar.
For \(B\), solving \((B-I)v=0\) gives \[ v= \begin{pmatrix} a\\b\\a \end{pmatrix}, \] so \[ \dim\ker(B-I)=2. \] For \(F\), \[ F-I= \begin{pmatrix} -1&1&1\\ 0&0&0\\ 1&0&-1 \end{pmatrix}. \] The equations \((F-I)v=0\) give \(x=z\) and \(y=0\), hence \[ \ker(F-I) = \RR \begin{pmatrix} 1\\0\\1 \end{pmatrix} \] and \[ \dim\ker(F-I)=1. \] The dimension of an eigenspace is invariant under similarity, so \(B\) and \(F\) are not similar.
6. No matrix in \(\{A,C,D,E\}\) is similar to \(B\) or \(F\).
By step 1, the two sets have different characteristic polynomials, and the characteristic polynomial is invariant under similarity.
7. Therefore the only distinct matrices in the list that are similarare\[ \boxed{C\sim E.} \]
Step 3 gives this similarity. Steps <1>4–<1>6 exclude every other pair.
8. Q.E.D.
Step 7 is the complete classification requested.