Linear Algebra

• orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. • transformation is defined in such a way that the first principal component has the largest possible variance and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components • resulting vectors are an uncorrelated orthogonal basis set. • PCA is sensitive to the relative scaling of the original variables.

Av = Lv, (L is lambda) v is eigen vector, lambda is eigenvalue

the singular-value decomposition of an mxn real or complex matrix M is a factorization of the form USigma V , where U is an mxm real or complex unitary matrix, Sigma is a mxn rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an n x n real or complex unitary matrix. The diagonal entries sigma_i of Sigma are known as the singular values of M