Stat 5101 Lecture Notes - School of Statistics

Appendix EEigenvalues andEigenvectorsE.1 Orthogonal and Orthonormal VectorsIf x and y are vectors of the same dimension, we say they are orthogonalif x ′ y = 0. Since the transpose of a matrix product is the product of thetransposes in reverse order, an equivalent condition is y ′ x = 0. Orthogonalityis the n-dimensional generalization of perpendicularity. In a sense, it says thattwo vectors make a right angle.The length or norm of a vector x =(x 1 ,...,x n ) is defined to be‖x‖ = √ ∑x ′ x = √ n x 2 i .i=1Squaring both sides gives‖x‖ 2 =n∑x 2 i ,which is one version of the Pythagorean theorem, as it appears in analyticgeometry.Orthogonal vectors give another generalization of the Pythagorean theorem.We say a set of vectors {x 1 ,...,x k } is orthogonal ifi=1x ′ ix j =0, i ≠ j. (E.1)231