Let-A-2-2-1-3-Find-a-non-singular-matrix-P-such-that-P-1-AP-is-a-diagonal-matrix-

Question Number 98250 by bobhans last updated on 12/Jun/20

Let A = (\begin{matrix} 2 2 \\ 1 3 \end{matrix}) . Find a non singular matrix

P such that P^{- 1} AP is a diagonal matrix .

Commented by john santu last updated on 12/Jun/20

find eigen vector \det (A - λ I) = 0

| \begin{matrix} 2 - λ 2 \\ 1 3 - λ \end{matrix} | = 0 \Rightarrow λ = 1; 4

Eigen vector for λ = 1

(A - I) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) \Rightarrow (\begin{matrix} 1 2 \\ 1 2 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix})

eigen - vector [\begin{matrix} - 2 \\ 1 \end{matrix}]

for λ = 4 \Rightarrow (\begin{matrix} - 2 2 \\ 1 - 1 \end{matrix}) [\begin{matrix} x \\ y \end{matrix}] = (\begin{matrix} 0 \\ 0 \end{matrix})

\to x - y = 0; eigen - vector = [\begin{matrix} 1 \\ 1 \end{matrix}]

∴ P = [\begin{matrix} - 2 1 \\ 1 1 \end{matrix}] such that

P^{- 1} AP = [\begin{matrix} 1 0 \\ 0 4 \end{matrix}]