Question

Consider the matrix \(A = \left[ {\begin{array}{*{20}{c}} {50}&{70}\\ {70}&{80} \end{array}} \right]\) whose eigenvectors corresponding to eigenvalues \({\lambda _1}\;and\;{\lambda _2}\;are\;{X_1} = \left[ {\begin{array}{*{20}{c}} {70}\\ {{\lambda _1} - 50} \end{array}} \right]and\;{X_2} = \left[ {\begin{array}{*{20}{c}} {{\lambda _2} - 80}\\ {70} \end{array}} \right]\) respectively. The value of \(x_1^T{x_2}\) is ________.

Answer 1

\(A = \left[ {\begin{array}{*{20}{c}} {50}&{70}\\ {70}&{80} \end{array}} \right]\)

Eigen values of A are λ₁ and λ₂

According to properties of eigen values,

Sum of Eigen values = Trace of matrix

= sum of diagonal Elements.

So, λ₁ + λ₂ = 80 + 50 = 130

Product of eigen values = Determinant of matrix

λ₁⋅λ₂ = (80 × 50) – (70 × 70) = -900

Now \({X_1} = \left[ {\begin{array}{*{20}{c}} {70}\\ {\lambda - 50} \end{array}} \right],\;{X_2} = \left[ {\begin{array}{*{20}{c}} {{\lambda _2} - 80}\\ {70} \end{array}} \right]\)

\(X_1^T{X_2} = \left[ {\begin{array}{*{20}{c}} {70}&{{\lambda _1} - 50} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\lambda _2} - 80}\\ {70} \end{array}} \right]\)

= 70 λ₂ – 5600 + 70λ₁ – 3500

= 70 (λ₁ + λ₂) – 9100 = 70 × 130 – 9100

= 9100 – 9100 = 0