Editing Definite matrix (section)

=== Further properties ===

# If <math>M</math> is a symmetric [[Toeplitz matrix]], i.e. the entries <math>m_{ij}</math> are given as a function of their absolute index differences: <math>m_{ij} = h(|i-j|),</math> and the ''strict'' inequality <math display="inline">\sum_{j \neq 0} \left|h(j)\right| < h(0)</math> holds, then <math>M</math> is ''strictly'' positive definite.
# Let <math>M > 0</math> and <math>N</math> Hermitian. If <math>MN + NM \ge 0</math> (resp., <math>MN + NM > 0</math>) then <math>N \ge 0</math> (resp., <math>N > 0</math>).<ref> {{Cite book
  | title=Positive Definite Matrices
  | last=Bhatia | first=Rajendra
  | publisher=Princeton University Press
  | year=2007
  | isbn=978-0-691-12918-1
  | location=Princeton, New Jersey
  | pages=8
 }}</ref>
# If <math>M > 0</math> is real, then there is a <math>\delta > 0</math> such that <math>M > \delta I,</math> where <math>I</math> is the [[identity matrix]].
# If <math>M_k</math> denotes the leading <math>k \times k</math> minor, <math>\det\left(M_k\right)/\det\left(M_{k-1}\right)</math> is the {{mvar|k}}th pivot during [[LU decomposition]].
# A matrix is negative definite if its {{mvar|k}}th order leading [[principal minor]] is negative when <math>k</math> is odd, and positive when <math>k</math> is even.
# If <math>M</math> is a real positive definite matrix, then there exists a positive real number <math>m</math> such that for every vector <math>\mathbf{v},</math> <math>\mathbf{v}^\mathsf{T} M\mathbf{v} \geq m\|\mathbf{v}\|_2^{2}.</math> 
# A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. It is however not enough to consider the leading principal minors only, as is checked on the diagonal matrix with entries {{math|0}} and {{math|−1&nbsp;.}}