Question

Matrix A is said to be involutory if A2 = I. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.

Answer 1

4 · 1/4 (I-0) = (A-0)∧2

see details in the graph

Explanation:

Matrix A is expressed in the form A∧2=I

To proof that Matrix A is both orthogonal and involutory, if and only if A is symmetric is shown by re-expressing that

A∧2=I in the standard form

4 · 1/4 (I-0) = (A-0)∧2

Re-expressing

A∧2 = I as a graphical element plotted on the graph

X∧2=I

The orthogonality is shown in the graphical plot displayed in the picture. Orthogonality expresses the mutually independent form of two vectors expressed in their perpendicularity.