Please answer with as much detail and show as much work as possible. True or Fal
ID: 3033226 • Letter: P
Question
Please answer with as much detail and show as much work as possible.
True or False. For all subquestions below, assume that A is an m times n real matrix. T F: Every eigenvalue of AA^T is nonnegative. TF:AA^T and A^T A have the same set of eigenvalues if not counting repeated zeros. T F: If A = U sigma V^T is a singular value decomposition of A. Then columns of U are eigenvectors of AA^T, columns of V are eigenvectors of A^T A. T F: If x, y elementof R^n are eigenvectors of A and are linearly independent, then x and y are orthogonal. T F: The principal components of a data set are orthogonal.Explanation / Answer
(i) T : if is an eigenvalue of AAT, then for some nonzero x,
x2=x,x=AATx, x=Ax,Ax0.
(ii) T : AAT and ATA have the same set of values if repeated zeros are not counted.
(iii) T : In calculating the SVD consists of finding the eigenvalues and eigenvectors of AAT and ATA. The eigenvectors of ATA make up the columns of V , the eigenvectors of AAT make up the columns of U.
(iv) F : One of the vectors may be the zero vector. This is True for non-zero orthogonal vectors.
(v) T : Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.