Shallow water equations

In mathematics, the Binomial Inverse Theorem is useful for expressing matrix inverses in different ways.

If A, U, B, V are matrices of sizes p×p, p×q, q×q, q×p, respectively, then

${(\mathbf{A}+\mathbf{U}\mathbf{B}\mathbf{V})}^{-1}={\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}\mathbf{U}\mathbf{B}{(\mathbf{B}+{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}\mathbf{U}\mathbf{B})}^{-1}{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}$

provided A and B + BVA⁻¹UB are nonsingular. Note that if B is invertible, the two B terms flanking the quantity inverse in the right-hand side can be replaced with (B⁻¹)⁻¹, which results in

${(\mathbf{A}+\mathbf{U}\mathbf{B}\mathbf{V})}^{-1}={\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}\mathbf{U}{({\mathbf{B}}^{-1}+{\mathbf{V}\mathbf{A}}^{-1}\mathbf{U})}^{-1}{\mathbf{V}\mathbf{A}}^{-1}.$

This is the matrix inversion lemma, which can also be derived using matrix blockwise inversion.

Verification

First notice that

$(\mathbf{A}+\mathbf{U}\mathbf{B}\mathbf{V}){\mathbf{A}}^{-1}\mathbf{U}\mathbf{B}=\mathbf{U}\mathbf{B}+{\mathbf{U}\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}\mathbf{U}\mathbf{B}=\mathbf{U}(\mathbf{B}+{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}\mathbf{U}\mathbf{B}).$

Now multiply the matrix we wish to invert by its alleged inverse

$(\mathbf{A}+\mathbf{U}\mathbf{B}\mathbf{V})({\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}\mathbf{U}\mathbf{B}{(\mathbf{B}+{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}\mathbf{U}\mathbf{B})}^{-1}{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1})$

$={\mathbf{I}}_p+{\mathbf{U}\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}-\mathbf{U}(\mathbf{B}+{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}\mathbf{U}\mathbf{B}){(\mathbf{B}+{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}\mathbf{U}\mathbf{B})}^{-1}{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}$

$={\mathbf{I}}_p+{\mathbf{U}\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}-{\mathbf{U}\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}={\mathbf{I}}_p$

which verifies that it is the inverse.

So we get that—if A⁻¹ and ${(\mathbf{B}+{\mathbf{B}\mathbf{V}\mathbf{A}}^{-1}\mathbf{U}\mathbf{B})}^{-1}$ exist, then ${(\mathbf{A}+\mathbf{U}\mathbf{B}\mathbf{V})}^{-1}$ exists and is given by the theorem above.

Special cases

If p = q and U = V = I_p is the identity matrix, then

${(\mathbf{A}+\mathbf{B})}^{-1}={\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}\mathbf{B}{(\mathbf{B}+{\mathbf{B}\mathbf{A}}^{-1}\mathbf{B})}^{-1}{\mathbf{B}\mathbf{A}}^{-1}.$

Remembering the identity

${\left(\mathbf{A}\mathbf{B}\right)}^{-1}={\mathbf{B}}^{-1}{\mathbf{A}}^{-1}.$

we can also express the previous equation in the simpler form as

${(\mathbf{A}+\mathbf{B})}^{-1}={\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}{(\mathbf{I}+\mathbf{B}{\mathbf{A}}^{-1})}^{-1}\mathbf{B}{\mathbf{A}}^{-1}.$

If B = I_q is the identity matrix and q = 1, then U is a column vector, written u, and V is a row vector, written v^T. Then the theorem implies

${(\mathbf{A}+{\mathbf{u}\mathbf{v}}^{\mathrm{T}})}^{-1}={\mathbf{A}}^{-1}-\frac{{\mathbf{A}}^{-1}{\mathbf{u}\mathbf{v}}^{\mathrm{T}}{\mathbf{A}}^{-1}}{1+{\mathbf{v}}^{\mathrm{T}}{\mathbf{A}}^{-1}\mathbf{u}}.$

This is useful if one has a matrix $A$ with a known inverse A⁻¹ and one needs to invert matrices of the form A+uv^T quickly.

If we set A = I_p and B = I_q, we get

${({\mathbf{I}}_p+\mathbf{U}\mathbf{V})}^{-1}={\mathbf{I}}_p-\mathbf{U}{({\mathbf{I}}_q+\mathbf{V}\mathbf{U})}^{-1}\mathbf{V}.$

In particular, if q = 1, then

${(\mathbf{I}+{\mathbf{u}\mathbf{v}}^{\mathrm{T}})}^{-1}=\mathbf{I}-\frac{{\mathbf{u}\mathbf{v}}^{\mathrm{T}}}{1+{\mathbf{v}}^{\mathrm{T}}\mathbf{u}}.$

See also

• Woodbury matrix identity
• Sherman-Morrison formula
• Invertible matrix
• Matrix determinant lemma
• For certain cases where A is singular and also Moore-Penrose pseudoinverse, see Kurt S. Riedel, A Sherman—Morrison—Woodbury Identity for Rank Augmenting Matrices with Application to Centering, SIAM Journal on Matrix Analysis and Applications, 13 (1992)659-662, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park. preprint Template:MR
• Moore-Penrose pseudoinverse#Updating the pseudoinverse