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

${(𝐀+𝐔𝐁𝐕)}^{-1}={𝐀}^{-1}-{𝐀}^{-1}𝐔𝐁{(𝐁+{𝐁𝐕𝐀}^{-1}𝐔𝐁)}^{-1}{𝐁𝐕𝐀}^{-1}$

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

${(𝐀+𝐔𝐁𝐕)}^{-1}={𝐀}^{-1}-{𝐀}^{-1}𝐔{({𝐁}^{-1}+{𝐕𝐀}^{-1}𝐔)}^{-1}{𝐕𝐀}^{-1}.$

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

Verification

First notice that

$(𝐀+𝐔𝐁𝐕){𝐀}^{-1}𝐔𝐁=𝐔𝐁+{𝐔𝐁𝐕𝐀}^{-1}𝐔𝐁=𝐔(𝐁+{𝐁𝐕𝐀}^{-1}𝐔𝐁).$

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

$(𝐀+𝐔𝐁𝐕)({𝐀}^{-1}-{𝐀}^{-1}𝐔𝐁{(𝐁+{𝐁𝐕𝐀}^{-1}𝐔𝐁)}^{-1}{𝐁𝐕𝐀}^{-1})$

$={𝐈}_p+{𝐔𝐁𝐕𝐀}^{-1}-𝐔(𝐁+{𝐁𝐕𝐀}^{-1}𝐔𝐁){(𝐁+{𝐁𝐕𝐀}^{-1}𝐔𝐁)}^{-1}{𝐁𝐕𝐀}^{-1}$

$={𝐈}_p+{𝐔𝐁𝐕𝐀}^{-1}-{𝐔𝐁𝐕𝐀}^{-1}={𝐈}_p$

which verifies that it is the inverse.

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

Special cases

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

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

Remembering the identity

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

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

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

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

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

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

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

In particular, if q = 1, then

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

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