Orthogonal Matrices and Quadratic Forms

Transposing is when the rows become columns(or columns become rows) in a matrix, https://chortle.ccsu.edu/VectorLessons/vmch13/vmch13_14.html.

(A+B)^T = A^T + B^T

(AB)^T = B^T*A^T

x^T*y = x dot y

if x is a column vector

x^T*x = IIxII^2 or length of x squared

http://mathworld.wolfram.com/OrthogonalMatrix.html

1) The matrix P is orthogonal if P^T*P = I or P^T = P^-1

2) A matrix is orthogonally diagonalisable if P^TAP = P^-1AP = D

A symmetric matrix is when A = A^T

3) If A is a symmetric matrix and v,w are the corresponding eigenvectors, v&w are orthogonal

Quadratic forms

x^TAx with x being column vectors of x1,x2 and A being a symmetric matrix.

x^TAx = ax1^2 + 2cx1x2 + bx2^2