- \( c_A(\lambda) = (\lambda + 1)^3 \), factors completely
-
\( \dim G_{-1}(A) = 3 \), which agrees with the algebraic multiplicity of \(\lambda = -1\) from the characteristic polynomial.
This generalized eigenspace is, by itself, a complete set of independent subspaces of \(\mathbb{R}^3\).
-
\( P = \begin{bmatrix}
3 & 1 & 1 \\
4 & 0 & 0 \\
0 & 4 & 0
\end{bmatrix} \)
\( P^{-1} A P = \begin{bmatrix}
-1 & 0 & -1 \\
0 & -1 & -1 \\
0 & 0 & -1
\end{bmatrix} \)
- \( c_A(\lambda) = (\lambda + 1)^2 (x - 1)^2 \), factors completely
-
\( \dim G_{-1}(A) = 2 \), which agrees with the algebraic multiplicity of \(\lambda = -1\) from the characteristic polynomial.
\( \dim G_1(A) = 2 \), which agrees with the algebraic multiplicity of \(\lambda = 1\) from the characteristic polynomial.
Since these two dimensions add to \(4\),
and we know that generalized eigenspaces from different eigenvalues are always independent,
these two generalized eigenspace together form a complete set of independent subspaces of \(\mathbb{R}^4\).
-
\( P = \begin{bmatrix}
1 & 1 & 5 & -1 \\
0 & 0 & 2 & 1 \\
0 & 1 & 2 & 0 \\
1 & 0 & 1 & 0
\end{bmatrix}\)
\( P^{-1} A P = \begin{bmatrix}
-1 & 1 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 2 \\
0 & 0 & 0 & 1
\end{bmatrix}\)