Consider the symmetric matrix

$$M = \begin{bmatrix} A & B \\ B^T & -C \end{bmatrix}$$

where $A$ (in $\cal{R}^{n\times n}$) and $C$ (in $\cal{R}^{m\times m}$) are positive semi-definite (symmetric) matrices (they are highly sparse).

My question is if there is an efficient way of checking if the Schur complement $S = A + BC^{-1}B^T$ is invertible by looking at the properties of $M$, $A$, $B$, $C$?

P.S: Inverting $C$ is costly and the inverse of a sparse matrix is a full matrix, something not desirable.

Thank you all.

Consider the symmetric matrix

$$M = \begin{bmatrix} A & B \\ B^T & -C \end{bmatrix}$$

where $A \in \cal{R}^{n \times n}$ and $C \in \cal{R}^{m\times m}$ are symmetric, positive semidefinite, highly sparse matrices.

Is there an efficient way of checking if the Schur complement

$$S = A + BC^{-1}B^T$$

is invertible by looking at the properties of matrices $M$, $A$, $B$, $C$?

Inverting $C$ is expensive and the inverse of a sparse matrix is a full matrix, which is undesirable.

Thank you all.

Consider the symmetric matrix

$$ M = \begin{bmatrix} A & B \\ B^T & -C \end{bmatrix} $$ , where $A$ (in $\cal{R}^{n\times n}$) and $C$ (in $\cal{R}^{m\times m}$) are positive semi-definite (symmetric) matrices (they are highly sparse).

My question is if there is an efficient way of checking if the Schur complement $S = A + BC^{-1}B^T$ is invertible by looking at the properties of $M$, $A$, $B$, $C$?

P.S: Inverting $C$ is costly and the inverse of a sparse matrix is a full matrix, something not desirable.

Thank you all.

Consider the symmetric matrix

$$M = \begin{bmatrix} A & B \\ B^T & -C \end{bmatrix}$$

where $A$ (in $\cal{R}^{n\times n}$) and $C$ (in $\cal{R}^{m\times m}$) are positive semi-definite (symmetric) matrices (they are highly sparse).

My question is if there is an efficient way of checking if the Schur complement $S = A + BC^{-1}B^T$ is invertible by looking at the properties of $M$, $A$, $B$, $C$?

P.S: Inverting $C$ is costly and the inverse of a sparse matrix is a full matrix, something not desirable.

Thank you all.