Necessary condition for definiteness of a block matrix

Question

Let a block matrix $$Y=\begin{bmatrix} A & F & G\\ F' & B & E\\ G' & E' & C\end{bmatrix}$$

Suppose that none of the matrices $A,B,C,E,F,G$ are equal $0$. In order to $Y$ 'be' positive definite one necessary condition is that $A,B$ and $C$ are positive definite?

Answer 1

Hint

Assume $A_{m\times m}, B_{n\times n}$ and $C_{p\times p}.$ Now, note that

$$(x_1,\cdots, x_m,0,\cdots, 0)Y(x_1,\cdots, x_m,0,\cdots, 0)^T=(x_1,\cdots, x_m,)A(x_1,\cdots, x_m)^T,$$

$$(\underbrace{0,\cdots,0}_{m},x_1,\cdots, x_n,\underbrace{0,\cdots, 0}_{p})Y(0,\cdots,0,x_1,\cdots, x_n,0,\cdots, 0)^T=(x_1,\cdots, x_n)B(x_1,\cdots, x_n)^T,$$ and

$$(0\cdots,0,x_1,\cdots, x_p)Y(0\cdots,0,x_1,\cdots, x_p)^T=(x_1,\cdots, x_p)C(x_1,\cdots, x_p)^T.$$

Thus the answer should be clear.