ORTHOGONALITY AND LEAST SQUARES

Orthogonal Vectors Norm Unit Vector Orthonormal Vectors Orthogonal Complement Orthogonal Projection Cauchy-Schwarz Inequality Angle Between Two Vectors Gram-Schmidt Process QR Factorialization Transpose Symmetric and Skew-symmetric Matrices Orthogonal Matrices Least Squares Solutions Matrix of an Orthogonal Projection

QR Factorialization How to Examples Exercise How to: QR Factorialization is an extension of the Gram-Schmidt Process. Here we take a matrix A, whose columns form a regular basis v1, ... , vM and we split it into the product of two other matrices, Q and R. Q has the property that its columns form an orthonormal basis for A, w1, ... , wM. R has the following properties:      R(1, 1) = norm(v1)      R(J, J) = norm(vJ - projVJminus1vJ) (J > 1)      R(I, J) = dot(wI, vJ) (I < J)      R(I, J) = 0 (I > J) To compute these two matrices using MATLAB simply input:      [Q, R] = qr(A)      Examples: Exercise: