Consider a $$3 \times 3$$ real symmetric matrix $$S$$ such that two of its eigen values are $$a \ne 0,$$ $$b\,\, \ne 0$$ with respective eigen vectors $$\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right],\left[ {\matrix{ {{y_1}} \cr {{y_2}} \cr {{y_3}} \cr } } \right].$$ If $$a\, \ne b$$ then $${x_1}{y_1} + {x_2}{y_2} + {x_3}{y_3}$$ equals

The eigen values of a symmetric matrix are all

For the matrix $$A = \left[ {\matrix{ 5 & 3 \cr 1 & 3 \cr } } \right],$$ ONE of the normalized eigen vectors is given as

Eigen values of a real symmetric matrix are always