Table 2 Matrix identities

I

\( {\left[\begin{array}{cc}\mathbf{A}& \mathbf{B}\\ {}{\mathbf{B}}^{\mathrm{T}}& \mathbf{C}\end{array}\right]}^{-1}=\left[\begin{array}{cc}{\left(\mathbf{A}-{\mathbf{B}\mathbf{C}}^{-1}{\mathbf{B}}^{\mathrm{T}}\right)}^{-1}& -{\left(\mathbf{A}-{\mathbf{B}\mathbf{C}}^{-1}{\mathbf{B}}^{\mathrm{T}}\right)}^{-1}{\mathbf{B}\mathbf{C}}^{-1}\\ {}-{\mathbf{C}}^{-1}{\mathbf{B}}^{\mathrm{T}}{\left(\mathbf{A}-{\mathbf{B}\mathbf{C}}^{-1}{\mathbf{B}}^{\mathrm{T}}\right)}^{-1}& {\left(\mathbf{C}-{\mathbf{B}}^{\mathrm{T}}{\mathbf{A}}^{-1}\mathbf{B}\right)}^{-1}\end{array}\right] \)

(A and C are symmetric matrices)

II

(A + BCD)⁻¹ = A⁻¹ − A⁻¹B(C⁻¹ + DA⁻¹B)⁻¹DA⁻¹

III

Π^⊥[A] = I − A(A^TA)⁻¹A^T when A is full column rank