the diagonal elements of a harmitian matrix are real.
the diagonal elements of a harmitian matrix are?
real
bullet::::- its leading principal minors are all positive: the "k"th leading principal minor of a matrix formula_2 is the determinant of its upper-left formula_199 sub-matrix. it turns out that a matrix is positive definite if and only if all these determinants are positive. this condition is known as sylvester's criterion, and provides an efficient test of positive definiteness of a symmetric real matrix. namely, the matrix is reduced to an upper triangular matrix by using elementary row operations, as in the first part of the gaussian elimination method, taking care to preserve the sign of its determinant during pivoting process. since the "k"th leading principal minor of a triangular matrix is the product of its diagonal elements up to row formula_200, sylvester's criterion is equivalent to checking whether its diagonal elements are all positive. this condition can be checked each time a new row formula_200 of the triangular matrix is obtained.
definiteness of a matrix
more generally, an matrix with "distinct nonzero eigenvalues" has 2 square roots. such a matrix, , has a decomposition where is the matrix whose columns are eigenvectors of and is the diagonal matrix whose diagonal elements are the corresponding eigenvalues . thus the square roots of are given by , where is any square root matrix of , which, for distinct eigenvalues, must be diagonal with diagonal elements equal to square roots of the diagonal elements of ; since there are two possible choices for a square root of each diagonal element of , there are 2 choices for the matrix .
square root of a matrix