Positive definite and block matrix

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?

• Every principal submatrix of a positive (semi)definite matrix is positive (semi)definite. Commented Apr 27, 2018 at 11:27
• Related post where it instead asks when is $M$ positive definite: math.stackexchange.com/questions/2280671/…. Commented Apr 27, 2018 at 11:30

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.