Definition: the matrix A is said to be positive if it is \(\displaystyle A_{ij} x^i x^j > 0\) for \(\displaystyle \forall x^i \neq 0^i\)
Theorem: \(\displaystyle H_p\) is the minor obtained by obliterating the n-p rows and the n-p columns from A. The matrix A is positive definite only if \(\displaystyle H_p > 0\) for \(\displaystyle \forall \; p = 1, \dots n\).
We demonstrate the theorem by induction on n. It is true for \(\displaystyle n = 1\). Suppose it is true for \(\displaystyle n - 1\) variables and prove it valid in the case of \(\displaystyle n\) variables.
If \(\displaystyle A_{11} > 0\) then we have
I don't understand the meaning of the last formula: what about \(\displaystyle \sum\limits_{i,j}^{2,\dots ,n} A_{ij} x^i x^j\) and the other terms? Can you help me?
Thanks in advances
Theorem: \(\displaystyle H_p\) is the minor obtained by obliterating the n-p rows and the n-p columns from A. The matrix A is positive definite only if \(\displaystyle H_p > 0\) for \(\displaystyle \forall \; p = 1, \dots n\).
We demonstrate the theorem by induction on n. It is true for \(\displaystyle n = 1\). Suppose it is true for \(\displaystyle n - 1\) variables and prove it valid in the case of \(\displaystyle n\) variables.
If \(\displaystyle A_{11} > 0\) then we have
I don't understand the meaning of the last formula: what about \(\displaystyle \sum\limits_{i,j}^{2,\dots ,n} A_{ij} x^i x^j\) and the other terms? Can you help me?
Thanks in advances