Suppose that the 3 x 3 matrix A {“version”:”1.1″,”math”:”\(…
Suppose that the 3 x 3 matrix A {“version”:”1.1″,”math”:”\(A\)”} has eigenvalues λ 1 = − 1 , λ 2 = 1 , λ 3 = 2 {“version”:”1.1″,”math”:”\(\lambda_1=-1, \lambda_2=1, \lambda_3=2\)”}, with corresponding eigenvectors v 1 = ( 0 , 5 , 3 ) t , v 2 = ( 2 , 0 , 1 ) t {“version”:”1.1″,”math”:”\(v_1=(0,5,3)^t, v_2=(2,0,1)^t\) “} and v 3 = ( 1 , − 1 , 0 ) t {“version”:”1.1″,”math”:”\(v_3=(1,-1,0)^t\)”}.If you diagonalize A {“version”:”1.1″,”math”:”\(A\)”} as A = P D P − 1 {“version”:”1.1″,”math”:”\(A=PDP^{-1}\)”} with P = ( 2 2 0 p 21 p 22 2 p 31 p 32 p 33 ) , D = ( 2 0 0 0 1 0 0 0 − 1 ) {“version”:”1.1″,”math”:”$$P=\begin{pmatrix} 2&2&0\\p_{21}&p_{22}& 2\\p_{31}&p_{32}&p_{33}\end{pmatrix}, \quad D=\begin{pmatrix} 2&0&0\\0&1& 0\\0&0&-1\end{pmatrix}$$”} then