An eigenvalue or characteristic value lor latent root) of a given Il X n matrix A =

Chapter 20, Problem 20.1.98

(choose chapter or problem)

If we take equal factors together and denote the numerically distinct eigenvalues of A by

\(\lambda_{1}, \cdots, \lambda_{r}(r \leqq n)\), then the product becomes

\(f(\lambda)=(-1)^{n}\left(\lambda-\lambda_{1}\right)^{m_{1}}\left(\lambda-\lambda_{2}\right)^{m_{2}} \cdots\left(\lambda-\lambda_{r}\right)^{m_{r}}\).

The exponent \(m_{j}\) is called the algebraic multiplicity of \(\lambda_{j}\). The maximum number of linearly independent eigenvectors corresponding to \(\lambda_{j}\); is called the geometric multiplicity of \(\lambda_{j}\). It is equal to or smaller than \(m_{j}\).

Text Transcription:

f(lambda)=(-a)^n(lambda-lambda_1)^m1(lambda-lambda_2)^m2…(lambda-lambda_r)^mr

lambda_1,...,lambda_r(r leqq n)

m_j

lambda_j