Definition: Let \( A \) be an \( n \times n \) matrix. A real number \( \lambda \) is called an eigenvalue of \( A \) if there exists a nonzero vector \( \mathbf{v} \) in \( \mathbb{R}^n \) for which \[ A\mathbf{v} = \lambda \mathbf{v} \] Such a (nonzero) vector \( \mathbf{v} \) is then called an eigenvector of \( A \) for the eigenvalue \( \lambda \)

\( A^n x = \lambda^n x \) for every \( n \) and \( (A + cI)x = (\lambda + c)x \) and \( A^{-1}x = x/\lambda \) if \( \lambda \neq 0 \)
\( (A - \lambda I)x = 0 \Rightarrow \) the determinant of \( A - \lambda I \) is zero: this equation produces \( n \) \(\lambda\)'s
Check \( \lambda \)'s by \( (\lambda_1)(\lambda_2)\cdots(\lambda_n) = \det A \) and \( \lambda_1 + \cdots + \lambda_n = \) diagonal sum \( a_{11} + \cdots + a_{nn} \)
Projections have \( \lambda = 1 \) and \( 0 \). Rotations have \( \lambda = e^{i\theta} \) and \( e^{-i\theta} \): complex numbers!

¹Linear Algebra. André Hensbergen & Nikolaas Verhulst. 2024. https://doi.org/10.59490/tb.88

²Introduction to Linear Algebra, Sixth Edition (2023). Gilbert Strang. ISBN : 978-17331466-7-8