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### 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

# If $$p\left( t \right) = {c_0} + {c_1}t + {c_2}{t^2} + ...... + {c_n}{t^n}$$, define $$p\left( A \right)$$ to be the matrix formed by replacing each power of $$t$$ in $$p\left( t \right)$$by the corresponding power of $$A$$ (with $${A^0} = I$$ ). That is,$$p\left( t \right) = {c_0} + {c_1}I + {c_2}{I^2} + ...... + {c_n}{I^n}$$Show that if $$\lambda$$ is an eigenvalue of A, then one eigenvalue of $$p\left( A \right)$$ is $$p\left( \lambda \right)$$.

It is proved that if $$\lambda$$ is an eigenvalue of $$\lambda$$, then one eigenvalue of $$p\left( A \right)$$ is $$p\left( \lambda \right)$$.

See the step by step solution

## Step 1: Definition of eigenvalue

Eigenvalues are a special set of scalars associated with a linear system of equations that are sometimes also known as characteristic roots, characteristic values, proper values, or latent roots.

## Step 2: Showing if is an eigenvalue of $$A$$, then one eigenvalue of $$p\left( A \right)$$ is $$p\left( \lambda \right)$$

Suppose that $$\lambda$$ is an eigenvalue of $$A$$ with the corresponding eigenvector$${\rm{x}}$$, that is, suppose that $$A{\rm{x}} = \lambda {\rm{x}},{\rm{x}} \ne 0$$.

Then,

$$\begin{array}{r}p\left( A \right){\rm{x}} = \left( {{c_0}I + {c_1}A + \ldots + {c_n}{A^n}} \right){\rm{x}}\\ = {c_0}{\rm{x}} + {c_1}A{\rm{x}} + \ldots + {c_n}{A^n}{\rm{x}}\end{array}$$

From exercise 4 we know that$${A^n}{\rm{x}} = {\lambda ^n}{\rm{x}}$$, so:

$$\begin{array}{c}p\left( A \right){\rm{x}} = {c_0}{\rm{x}} + {c_1}\lambda {\rm{x}} + \ldots + {c_n}{\lambda ^n}{\rm{x}}\\ = \left( {{c_0} + {c_1}\lambda + \ldots + {c_n}{\lambda ^n}} \right){\rm{x}}\\ = p(\lambda ){\rm{x}}\end{array}$$.

Hence it is proved that $$\lambda$$ is an eigenvalue of $$\lambda$$, then one eigenvalue of $$p\left( A \right)$$ is $$p\left( \lambda \right)$$.