Prove that if \(A\) and \(B\) are \(3 \times 3\) matrices over the field \(F\), a necessary and sufficient condition that \(A\) and \(B\) be similar over \(F\) is that they have the same characteristic polynomial and the same minimal polynomial. Give an example which shows that this is false for \(4 \times 4\) matrices.
Solution
1. (\(\Rightarrow\)) If \(A\) and \(B\) are similar, they have the same characteristic and minimal polynomials.
similar matrices have the same characteristic polynomial and the same minimal polynomial (both are similarity invariants).
2. (\(\Leftarrow\)) For \(3 \times 3\) matrices, the characteristic and minimal polynomials together determine the rational canonical form.
2.1. The rational canonical form is determined by the invariant factors \(d_1 \mid d_2 \mid \cdots \mid d_k\) with \(d_1 \cdots d_k = \chi\) (the characteristic polynomial) and \(d_k = m\) (the minimal polynomial).
the invariant factors determine the rational canonical form, and their product is the characteristic polynomial while the largest is the minimal polynomial.
2.2. For a \(3 \times 3\) matrix, \(\deg \chi = 3\), so the invariant factors have degrees summing to \(3\); the minimal polynomial \(m\) is the largest invariant factor.
degree count.
2.3. Given \(\chi\) and \(m\) (with \(m \mid \chi\)), the invariant factors are uniquely determined: if \(\deg m = 3\), then the single invariant factor is \(m = \chi\); if \(\deg m = 2\), then the invariant factors are \(d_1 = \chi/m\) (degree \(1\)) and \(d_2 = m\) (degree \(2\)); if \(\deg m = 1\), then \(m = x - \lambda\) and the invariant factors are three linear factors (determined by \(\chi\)).
enumerate the possible degree decompositions of \(3\).
2.4. Hence \(A\) and \(B\) have the same invariant factors, so the same rational canonical form, so they are similar.
3.
3. Counterexample for \(4 \times 4\): let \(A\) have invariant factors \((x, x, x^2)\) and \(B\) have invariant factors \((x^2, x^2)\).
3.1. Both have characteristic polynomial \(x^4\) and minimal polynomial \(x^2\).
the product of invariant factors is \(x^4\) in both cases, and the largest invariant factor is \(x^2\) in both cases.
3.2. But \(A\) and \(B\) are not similar.
they have different invariant factors (\((x, x, x^2)\) versus \((x^2, x^2)\)), hence different rational canonical forms.
4. Q.E.D.
1, 2, and 3.