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If matrix is Positive definite then its ALL Eigen Values > 0. this Implies that its quadratic form will be strictly positive.Let suppose A is such Matrix then
XTAX >0 .
XT means X tranpose. So then we can write the above in ax2+bx+C. this form is widely used in Statistical Signal processing to describe the Gaussian distribution of Multidimensional Random Variables
Every positive definite matrix is invertible and its inverse is also positive definite. If M is positive definite and r > 0 is a real number, then rM is positive definite. If M and N are positive definite, then the sum M + N and the products MNM and NMN are also positive definite; and if MN = NM, then MN is also positive definite. Every positive definite matrix M, has at least one square root matrix N such that N2 = M. In fact, M may have infinitely many square roots, but exactly one positive definite square root.
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