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Suppose a $n \times n$ symmetric matrix $\mathbf{M}$ is positive definite. Its block matrix form is written as follow. \begin{align} \mathbf{M} \; = \; \begin{pmatrix} \mathbf{A} & \mathbf{B}\\ \mathbf{B}^{T} & \mathbf{D} \end{pmatrix} \end{align}

Under what conditions can we say that the square matrices $\mathbf{A}$ or $\mathbf{D}$ are also positive definite?

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As Algebraic Pavel stated correctly:

Every principal submatrix of a positive (semi)definite matrix is positive (semi)definite.

Here some elaborate statements:

If $A\in\mathbb{R}^{n\times n}$ is positive definite, then all of its principal submatrices $a_{1:m,1:m}$ ($m=1,\dots, n$) are positive definite. If $A$ is positive semi-definite, then all of its principal submatrices $a_{1:m,1:m}$ ($m=1,\dots, n$) are positive semi-definite.

This also works for negative (semi)-definite matrices, by simply multiplying the matrix by -1, i.e.

If $-A\in\mathbb{R}^{n\times n}$ is positive definite, then all of its principal submatrices $-a_{1:m,1:m}$ ($m=1,\dots, n$) are positive definite. If $-A$ is positive semidefinite, then all of its principal submatrices $-a_{1:m,1:m}$ ($m=1,\dots, n$) are positive semidefinite.

For a reference, see Observation 7.1.2 from Matrix Analysis (Horn, Johnson), 2nd edition.

See also Sylvester's criterion, which is a similar statement regarding principle minors.

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