The identity matrix \(I_3\) is: \[ I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
The eigenvalues of \(I_3\) are all 1 since \(I_3 x = x\) for all vectors \(x\) in the space. Thus, the characteristic polynomial is \((1-\lambda)^3 = 0\) and it has a root \(\lambda = 1\) with algebraic multiplicity 3.
The eigenspace for \(\lambda = 1\) is the entire space \(\mathbb{R}^3\), and therefore the geometric multiplicity of \(\lambda = 1\) is also 3. The eigenvectors are any non-zero vector in \(\mathbb{R}^3\). A typical set of eigenvectors is the standard basis for \(\mathbb{R}^3\): \[ \mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \]
Each of these is an eigenvector of \(I_3\) corresponding to the eigenvalue \(\lambda = 1\).
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