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Table 2 Matrix identities

From: On the use of calibration emitters for TDOA source localization in the presence of synchronization clock bias and sensor location errors

Serial number Matrix identity
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(ATA)−1AT when A is full column rank