Explained: "Find a basis for the eigenspace corresponding to..."

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dictetzqh

Answered question

2021-11-21

Find a basis for the eigenspace corresponding to each listed eigenvalue.
$A = [\begin{matrix} 5 & 0 \\ 2 & 1 \end{matrix}], λ = 1, 5$

Answer & Explanation

Florence Pittman

Beginner2021-11-22Added 15 answers

We first solve the system to obtain the foundation for the eigenspace. $(A - λ l) x = 0$
For $λ = 1, A - l =$
$[\begin{matrix} 5 - 1 & 0 \\ 2 & 1 - 1 \end{matrix}]$
$[\begin{matrix} 4 & 0 \\ 2 & 0 \end{matrix}]$
The augment matrix of $(A - l) x = 0$ is
$[\begin{matrix} 4 & 0 & 0 \\ 2 & 0 & 0 \end{matrix}]$
With $x_{1} = 0$ and $x_{2}$ is unrestricted, the answer may be expressed as follows:
$x = x_{2}$
$[\begin{matrix} 0 \\ 1 \end{matrix}]$
So $[\begin{matrix} 0 \\ 1 \end{matrix}]$
is the foundation of the eigenspace.
For $λ = 5, A - 5 l =$
$[\begin{matrix} 5 - 5 & 0 \\ 2 & 1 - 5 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 2 & - 4 \end{matrix}]$
The augmented matrix of (A-5l)x=0 is
$[\begin{matrix} 0 & 0 & 0 \\ 2 & - 4 & 0 \end{matrix}]$
That leads to $2 x_{1} - 4 x_{2} = 0 \to x_{1} = 2 x_{2}$ . The answer may be written as follows:
$x = x_{2}$
$[\begin{matrix} 2 \\ 1 \end{matrix}]$
So
$[\begin{matrix} 2 \\ 1 \end{matrix}]$
is the foundation of the eigenspace.

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