Parallel Computing for Data Science

290 APPENDIX A. MATRIX REVIEW

• If A is symmetric and real, then it is diagonalizable, i.e., there exists

an orthogonal matrix U such that

U

0

AU = D (A.20)

for a diagonal matrix D. The elements of D are the eigenvalues of A,

and the columns of U are the eigenvectors of A.

A different sufficient condition for A.20 is that the eigenvalues of A

are distinct. In this case, U will not necessarily be orthogonal.

By the way, this latter sufficient condition shows that “most” square

matrices are diagonalizable, if we treat their entries as continuous ran-

dom variables. Under such a circumstance, the probability of having

repeated eigenvalues would be 0.

A.7 Matrix