Consider a square matrix A. The matrix B is generated by deleting the k-th row and k-th column of A. Is it true to say if A is negative definite then B is negative definite too?
2026-04-13 17:27:07.1776101227
The negative definiteness by deleting row and column
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Yes, it's true. This is because $A$ is negative-definite iff
$$\tag{1}\forall X \in \mathbb{R^n} \ \ X^TAX \le 0.$$
Deleting the $k$th line and column amounts to restrict (1) to vectors $X$ having a $0$ value as their $k$th coordinate, thus still true.