4.1.1 Preliminaries

Let S ^m be the space of symmetric m m matrices and M ^m,n be the space of rectangular m n matrices with real entries. From the viewpoint of their linear structure (i.e., the operations of addition and multiplication by reals), S ^m is just the arithmetic linear space R ^m(m+1)/2 of dimension by arranging the elements of a symmetric m m matrix X in a single column, say, in row-by-row order, you get a usual m ²-dimensional column vector; multiplication of a matrix by a real and addition of matrices correspond to the same operations with the representing vector(s). When X runs through S ^m, the vector representing X runs through m( m + 1)/2-dimensional subspace of R ^{m 2} consisting of vectors satisfying the symmetry condition-the coordinates coming from symmetric to each other pairs of entries in X are equal to each other. Similarly, M ^m,n as a linear space is just R ^m,n, and it is natural to equip M ^m,n with the inner product defined as the usual inner product of the vectors representing the matrices:

Here Tr stands for the trace-the sum of diagonal elements of a (square) matrix. With this inner...