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I am given a $J \times J$ symmetric matrix $\Sigma$. For a subset of $\{1, ..., J\}$, $v$, let $\Sigma_{v}$ denote the associated square submatrix. I am in need of an efficient scheme for inverting $\Sigma_v$ for potentially any subset $v$. Essentially, I will have to invert many different submatricies $\Sigma_v$, but I won't know which ones I need to invert in advance of running some program; I would rather invest in a good matrix decomposition at the outset, if one exists (or otherwise get whatever information necessary, if not a decomposition).

I've messed around with the eigen decomposition a little bit, but wasn't able to coerce the inverse of a submatrix out of it.

UPDATE: Apparently the term for the type of submatricies I want to invert is "principal submatrix". I'm wondering if I can't make some progress on this via the Cholesky decomposition. Suppose I'm content to calculate $\Sigma_{jj} ^ {-1}$ for any $j \in \{1, 2, ..., J\}$, where $\Sigma_{jj}$ denotes the submatrix obtained by deleting rows/columns greater than $j$. Write $\Sigma = LL^T$ and let $Q = L^{-1}$. Write $$ L = \begin{pmatrix}L_1 & \mathbf 0 \\ B & L_2\end{pmatrix}, Q = \begin{pmatrix}Q_1 & \mathbf 0 \\ C & Q_2\end{pmatrix} $$ where $L_1$ and $Q_1$ and $j \times j$. It follows that $\Sigma_{jj} = L_1 L_1^T$ and $Q_1 = L_1 ^{-1}$ so that $\Sigma_{jj} ^{-1} = Q_1^T Q_1$. So, once I have the Cholesky decomposition I have the inverses of the leading principal submatricies. This doesn't solve the problem as stated since I may need to deal with other principal submatricies, but it should be a useful partial solution.

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Would this be useful? It shows a way of updating the inverse when small changes are made to the matrix: alglib.net/matrixops/general/invupdate.php –  Bitwise Oct 6 '12 at 0:21
Assuming your matrix is real matrix (all elements are real), you are guaranteed by spectral theorem that your matrix $A = QDQ^{T}$ where $D$ is diagonal and $Q$ is orthogonal. Getting this $Q$ is algorithmically Gram-Schimdt process (so rather simple). –  s.b Oct 6 '12 at 0:22
@s.b yes, I'm aware; if that helps me invert the submatricies of $\Sigma$ (your $A$) then please tell me how, but as I said in my question I've already tried to figure out how to use the eigen decomposition to do it. –  guy Oct 6 '12 at 0:54
> I can tell you that there is a rank one update formula for finding > the inverse of a submatrix of one less dimension, which is to say the > matrix attained after deletion of any desired row and any desired > column. If you think it would be useful I could recite it for you > after I look it up exactly. I'm also interested in inverting this kind of matrices. If you could post the answer i would be glad. Thanks in advance, P –  user66612 Mar 13 '13 at 18:48
@user66612: I would be very happy to see it. –  Gil Mar 20 '13 at 12:15

3 Answers 3

The solution adam W proposed generalizes to rank-$n$ updates; it is not necessary to arrive by repeated rank-1 updates. In his derivation, we can consider $h$ a matrix instead of a scalar, and replace the bottom-right-hand 1 with an identity matrix and $\frac{1}{h}$ with $h^{-1}$. It still works.

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If we note $\Sigma_j = \Sigma_{[1..j-1]\cup[j+1..J]}$ the matrix $\Sigma$ with deletion of the $j$th row and column, the formulae for computing the updated Cholesky decomposition are : $$\Sigma = \begin{bmatrix} A_{11}&A_{12}&A_{13}\\A_{12}^T&A_{22}&A_{23}\\A_{13}^T&A_{23}^T&A_{33} \end{bmatrix} $$ $$L = \begin{bmatrix} L_{11}&L_{12}&L_{13}\\0&L_{22}&L_{23}\\0&0&L_{33} \end{bmatrix} $$

$$\Sigma_j = \begin{bmatrix} A_{11}&A_{13}\\A_{13}^T&A_{33} \end{bmatrix} $$

$$L_j = \begin{bmatrix} L_{11}&L_{13}\\0&\mathsf{cholupdate}(L_{33},L_{23})\end{bmatrix} $$ where $\mathsf{cholupdate}(A,B) = \mathsf{chol}(A^TA+B^TB)$. These computations perform in $O(n^2)$, so if you have $K$ deletions, the computation cost is $O(Kn^2)$.

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I imagine the best you can do is only one rank at a time building up or down to attain the desired sub-matrix portion. To reduce the dimension by one the simple formula is

$$\mathbf{A}^{-1} = E - \frac{\mathbf{f}\mathbf{g}^T}{h}$$

To see this, use the known inverse of the original and larger dimension matrix $$\pmatrix{\mathbf A & \mathbf b\\ \mathbf c^T & d}^{-1} = \pmatrix{\mathbf E & \mathbf f\\ \mathbf g^T & h}$$

to have $$\pmatrix{\mathbf E & \mathbf f\\ \mathbf g^T & h}\pmatrix{\mathbf A & \mathbf b\\ \mathbf c^T & d} = \pmatrix{ \mathbf I & \mathbf 0 \\ \mathbf 0^T & 1}$$

Now to find the quantity $A^{-1}$, simply left multiply the equation with $$\pmatrix{\mathbf{I} & -\mathbf{f}\frac{1}{h} \\ \mathbf{0}^T & 1}$$ giving $$ \pmatrix{\mathbf{E}- \mathbf{f}\frac{1}{h}\mathbf{g}^T & \mathbf{0} \\ \mathbf{g}^T & h}\pmatrix{\mathbf{A} & \mathbf{b} \\ \mathbf{c}^T & d} = \pmatrix{\mathbf{I} & -\mathbf{f}\frac{1}{h} \\ \mathbf{0}^T & 1}$$ The upper left portion of this matrix equation is $$\left( \mathbf{E} - \mathbf{f}\frac{1}{h}\mathbf{g}^T\right)\mathbf{A} = \mathbf{I}$$ and shows the formula.

To go the other direction (adding a row and column) you can use what is called the bordering method as described in this answer

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Thanks (+1). The point of looking into the eigen decomposition was that I could perhaps do it up front if it gave made submatrix inversion as trivial as it makes the full matrix inversion, but it doesn't appear to. The logic was that, while it is harder, it would save time in the long run because all the work would be done up front at once. –  guy Oct 6 '12 at 1:52
Yes, unfortunately (or fortunately else things might not be as interesting) eigenspace spectrums do not have easy update formulas. –  adam W Oct 6 '12 at 2:02
Also, having the spectrum is the same as having a matrix with inverse, that is what the left and right eigenvectors are after all. So this method could be used on those vectors if desired. I think :) –  adam W Oct 6 '12 at 2:10
Could you please recite the rank one update formula for finding the inverse? That would be extremely helpful for me. –  Gil Mar 20 '13 at 15:00
@Gil That would be the formula already given: $(A + uv^T)^{-1}=A^{-1}-\frac{(A^{-1}u)(v^TA^{-1})}{1 + v^TA^{-1}u}$. If you are referring to my "downdate" to lesser dimension as described in this answer, apply that formula to $E(\mathbf I-\mathbf {bg}^T)^{-1}$. So use $A=I$, $u=-\mathbf{b}$, and $v^\top = \mathbf{g}^\top$. When simplified, it becomes a rank one update of $\mathbf{E}$ –  adam W Mar 20 '13 at 17:48

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