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Q7.4-25E

Expert-verified
Found in: Page 267

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Consider an invertible n × n matrix A such that the zero state is a stable equilibrium of the dynamical system $\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{\left(}}{\mathbf{t}}{\mathbf{+}}{\mathbf{1}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{A}}\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{\left(}}{\mathbf{t}}{\mathbf{\right)}}$ What can you say about the stability of the systems$\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{\left(}}{\mathbf{t}}{\mathbf{+}}{\mathbf{1}}{\mathbf{\right)}}{\mathbf{=}}{{\mathbf{A}}}^{\mathbf{-}\mathbf{1}}\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{\left(}}{\mathbf{t}}{\mathbf{\right)}}$

The given value is unstable

See the step by step solution

## Definition of eigenvalue

An Eigenvalue is a scalar of linear operators for which there exists a non-zero vector. This property is equivalent to an Eigenvector.