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

Gram-Schmidt Process How to Examples Exercise How to: The Gram-Schmidt process takes a regular basis of a subspace V and constructs with it an orthonormal basis. So, we are given a regular basis v1, v2, ... , vM, and we want to find w1, w2, ... , wM, an orthonormal basis. Finding w1 is simple (unlike the others), it is simply v1 as a unit vector:      w1 = v1/norm(v1) Finding the other vectors is not as simple and requires an additional computation. To find wJ, we have to find the orthogonal projection of vJ onto a subspace of V which is the span of all the vectors before vJ (that is v1, v2, ... vJminus1). We'll call this subspace VJminus1 (since we can't use the '-' symbol in the name of variables). So, the orthogonal projection, projVJminus1vJ of vJ onto VJminus1 is given by:      projVJminus1vJ = dot(w1, vJ)*w1 + dot(w2, vJ)*w2 + ... + dot(wJminus1, vJ)*wJminus1 Now, we can use this to calculate wJ:      wJ = (vJ - projVJminus1vJ)/norm(vJ - projVJminus1vJ) This process can be very exhausting...luckily MATLAB simplifies it for us with QR-Factorialization.      Examples: Exercise: