Bert's blog

## Numerically inverting a matrix

Suppose you have a square matrix, like the one below:

$$A = \begin{pmatrix} 0 & 5 & 5 \\ 2 & 9 & 0 \\ 6 & 8 & 8 \end{pmatrix},$$

and that you want to compute the inverse of this matrix (numerically). The result should be (thanks WolframAlpha!):

$$A^{-1} = \begin{pmatrix} -\frac{4}{15} & 0 & \frac{1}{6} \\ \frac{8}{135} & \frac{1}{9} & -\frac{1}{27} \\ \frac{19}{135} & -\frac{1}{9} & \frac{1}{27} \end{pmatrix}$$

A first option to invert this matrix is to realise that this is a $3\times{}3$ matrix, for which Wikipedia lists a special formula. The problem of this approach is that it is not very general.

More generally, Wikipedia lists straightforward inversion algorithms that can be readily executed for matrices of any size (they obviously need to be square to be invertible). These involve a lot of operations, and require a lot of intermediate variables to be stored. Furthermore, these algorithms are not very stable against round off error; if the matrix contains elements with significantly different sizes, then accumulated round off error can lead to a significant loss of accuracy, or to numerical issues that cause the inversion algorithm to fail altogether, even if the matrix is strictly speaking invertible.

Fortunately, there is an elegant matrix inversion algorithm that does not suffer from these issues, and that furthermore can almost be executed in place, i.e. intermediate steps are mostly stored within the matrix itself, and only limited additional memory is required.

# PLU decomposition and matrix inversion

The basic idea of this better matrix inversion algorithm is to perform a so called PLU decomposition of the original matrix. This means rewriting the matrix as

Here, $P$, $L$ and $U$ are all special matrices, that is, $P$ is a so called permutation matrix (a unit matrix of which some of the rows/columns have been swapped), $L$ is a lower triangular matrix with a unit diagonal (a matrix with $1$s along the diagonal and all elements above and to the right of the diagonal $0$), and $U$ is an upper diagonal matrix (I will let you guess what that means).

A possible PLU decomposition for $A$ (this decomposition is not unique) is:

$$A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \times{} \begin{pmatrix} 1 & 0 & 0 \\ \frac{1}{3} & 1 & 0 \\ 0 & \frac{15}{19} & 1 \end{pmatrix} \times{} \begin{pmatrix} 6 & 8 & 8 \\ 0 & \frac{19}{3} & -\frac{8}{3} \\ 0 & 0 & \frac{135}{19} \end{pmatrix}.$$

I will explain later how to obtain this decomposition. Note that the PLU decomposition is a variant of the so called LU decomposition, where the matrix is simply decomposed in a lower and upper triangular matrix, without the permutation matrix. When the permutation matrix is used, we call this an LU decomposition with partial pivoting. I will come back to this point later, as it is very important for the stability of the matrix inversion algorithm.

Once the matrix has been decomposed, inverting it is done by inverting (some) of its factors:

There are straightforward algorithms to invert triangular matrices (I will show one later), and the inverse of a permutation matrix is simply its transpose. So using the PLU decomposition, computing the inverse matrix is straightforward:

$$A^{-1} = \begin{pmatrix} \frac{1}{6} & -\frac{4}{19} & -\frac{4}{15} \\ 0 & \frac{3}{19} & \frac{8}{135} \\ 0 & 0 & \frac{19}{135} \end{pmatrix} \times{} \begin{pmatrix} 1 & 0 & 0 \\ -\frac{1}{3} & 1 & 0 \\ \frac{5}{19} & -\frac{15}{19} & 1 \end{pmatrix} \times{} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \\ = \begin{pmatrix} \frac{1}{6} & 0 & -\frac{4}{15} \\ -\frac{1}{27} & \frac{1}{9} & \frac{8}{135} \\ \frac{1}{27} & -\frac{1}{9} & \frac{19}{135} \end{pmatrix} \times{} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$

You can easily see that the permutation matrix (or its transpose) now leads to column swaps in the matrix that eventually lead to the inverse matrix WolframAlpha found.

However, an even more practical algorithm to compute the inverse is given by rewriting the above equations as

and solving for $\left( P^{-1} \times{} A \right)^{-1}$. This algorithm is better, as it avoids having to compute the inverse of $L$, and also avoids the need to store intermediate matrices. Once you have computed $\left( P^{-1} \times{} A \right)^{-1}$, the inverse matrix can be found by multiplying it with the permutation matrix, as before:

# A PLU inversion algorithm

To find the PLU decomposition, we can use the Doolittle algorithm. This algorithm can compute $L$ and $U$ and store them in the original matrix, and only requires one additional array with a size that corresponds to the size of the matrix in one dimension (3 in our example) that is used to store the permutation matrix (in a compact form). The resulting matrix will contain the upper triangular and diagonal elements from $U$, and the lower triangular elements from $L$. We don’t need to store the diagonal elements of $L$, since we know they are all $1$.

The algorithm works as follows: we traverse the rows of $A$, and for each row $i$ try to find the element $A_{ji}$ (column $i$ in row $j$, with $j \geq{} i$) with the largest absolute value. The index $j$ is called the pivot. We record that index in an auxiliary pivot array. Once we have found and recorded the pivot, we swap rows $i$ and $j$ (note that $i$ and $j$ could be equal, in which case we don’t do anything).

The next step is to use row $i$ (this will be row $j$, but since we swapped rows, this row will now be on row $i$ in the original matrix) to eliminate all rows $j > i$ in column $i$ (i.e. make these elements $0$). We can do this by multiplying the top row with $\frac{A_{ji}}{A_{ii}}$ and subtracting this from the original row. This is allowed, provided that we multiply the matrix with a transformation matrix that happens to be a lower triangular matrix. After we have done this for all rows, it will turn out that all these transformation matrices multiplied together give us $L$, and the elements $L_{ji}$ will simply be the factors $\frac{A_{ji}}{A{ii}}$ (you can check this if you really want).

So in practice, all we need to do is:

1. Divide all elements in column $i$ below row $i$ by $A_{ii}$; this gives us the elements of $L$ in that column.
2. Subtract this factor times row $i$ from all columns $j > i$ in all rows $k > i$. This does the elimination for the part of the matrix that does not vanish after the elimination (and will give us the elements of $U$).

For our example matrix, we get the following (for clarity, I have indicated the pivot array in an additional column):

$$\begin{pmatrix} 0 & 5 & 5 & | & ? \\ 2 & 9 & 0 & | & ? \\ 6 & 8 & 8 & | & ? \end{pmatrix} \rightarrow{} \begin{pmatrix} 6 & 8 & 8 & | & 3 \\ 2 & 9 & 0 & | & ? \\ 0 & 5 & 5 & | & ? \end{pmatrix} \rightarrow{} \begin{pmatrix} 6 & 8 & 8 & | & 3 \\ \frac{1}{3} & \frac{19}{3} & -\frac{8}{3} & | & ? \\ 0 & 5 & 5 & | & ? \end{pmatrix} \\ \rightarrow{} \begin{pmatrix} 6 & 8 & 8 & | & 3 \\ \frac{1}{3} & \frac{19}{3} & -\frac{8}{3} & | & 2 \\ 0 & 5 & 5 & | & ? \end{pmatrix} \rightarrow{} \begin{pmatrix} 6 & 8 & 8 & | & 3 \\ \frac{1}{3} & \frac{19}{3} & -\frac{8}{3} & | & 2 \\ 0 & \frac{15}{19} & \frac{135}{19} & | & ? \end{pmatrix} \rightarrow{} \begin{pmatrix} 6 & 8 & 8 & | & 3 \\ \frac{1}{3} & \frac{19}{3} & -\frac{8}{3} & | & 2 \\ 0 & \frac{15}{19} & \frac{135}{19} & | & 3 \end{pmatrix}$$

You can recognise the $L$ and $U$ matrices I showed before.

Inverting the upper diagonal matrix $U$ is reasonably straightforward. First of all, it can be shown that the inverse of an upper/lower triangular matrix is always an upper/lower triangular matrix too. This immediately means that the diagonal elements of the inverted matrix are simply the inverse of the diagonal elements of the original matrix. If we denote the elements of the original matrix $U$ with upper case $U_{ij}$, and the elements of the inverse matrix $U^{-1}$ with lower case $u_{ij}$, then we need to solve the following equation:

$$\begin{pmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{pmatrix} \times{} \begin{pmatrix} U_{11} & U_{12} & U_{13} \\ 0 & U_{22} & U_{23} \\ 0 & 0 & U_{33} \end{pmatrix} \\= \begin{pmatrix} u_{11} U_{11} & u_{11} U_{12} + u_{12} U_{22} & u_{11} U_{13} + u_{12} U_{23} + u_{13} U_{33} \\ 0 & u_{22} U_{22} & u_{22} U_{23} + u_{23} U_{33} \\ 0 & 0 & u_{33} U_{33} \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix},$$

this is simply the definition of the inverse matrix.

You can see a pattern emerge that allows you to solve for the $u_{ij}$ recursively, starting from the top row and working down (a so called forward substitution). This algorithm only requires multiplication and subtraction, and a single division per element that only involves the diagonal elements of $U$ (this will be important later). Mathematically, we can write down the following formula for the elements $u_{ij}$ ($j > i$):

You can check that a forward substitution algorithm can be used in place, i.e. the $u_{ik}$ and $U_{kj}$ in this formula are all elements of the corresponding separate matrices (despite being stored in the same matrix), as long as you (a) store $u_{ij}$ in a temporary variable during the summation, and (b) traverse the matrix from top to bottom (from small to large $i$), and within a row from left to right (from small to large $j$). I will leave it up to you to check that this indeed yields the same inverse matrix $U^{-1}$ as I showed before.

The next step is to solve the equation $\left(P^{-1} \times{} A \right)^{-1} \times{} L = U^{-1}$. If we denote the elements of the matrix $\left(P^{-1} \times{} A\right)^{-1}$ with $a_{ij}$, the elements of $L$ with $L_{ij}$, and the elements of $U^{-1}$ with $u_{ij}$ as before, we get

$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \times{} \begin{pmatrix} 1 & 0 & 0 \\ L_{21} & 1 & 0 \\ L_{31} & L_{32} & 1 \end{pmatrix} \\ = \begin{pmatrix} a_{11} + a_{12} L_{21} + a_{13} L_{31} & a_{12} + a_{13} L_{32} & a_{13} \\ a_{21} + a_{22} L_{21} + a_{23} L_{31} & a_{22} + a_{23} L_{32} & a_{23} \\ a_{31} + a_{32} L_{21} + a_{33} L_{31} & a_{32} + a_{33} L_{32} & a_{33} \end{pmatrix} = \begin{pmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{pmatrix}$$

If you look carefully, you can see a clear pattern: the last column of the matrix $\left(P^{-1} \times{} A\right)^{-1}$ simply equals the last column of $U^{-1}$, while the other columns can be computed using only values from $L$, $U$ and the columns that were already computed, if we traverse the columns from right to left. Again, we can use an in place algorithm, as long as we make sure to store the new column in an intermediate array while it is being computed, as the calculation for each element of the new column requires all elements of $L$ for that same column, and we don’t want to overwrite them before we finished the column.

Mathematically, this backward substitution algorithm can be expressed as follows:

If we were to apply this algorithm to the intermediate matrix, we would end up with the same intermediate matrix I showed before, and we also still have the pivot array:

$$\begin{pmatrix} \frac{1}{6} & 0 & -\frac{4}{15} & | & 3 \\ -\frac{1}{27} & \frac{1}{9} & \frac{8}{135} & | & 2 \\ \frac{1}{27} & -\frac{1}{9} & \frac{19}{135} & | & 3 \end{pmatrix}$$

The only thing left to do is use the pivot array to rearrange the columns. Since mathematically we have to use the inverse of the permutation matrix, we will need to use a backwards algorithm, were we start from the last column, and then swap that column with the column with the index given by the last element of the pivot array, and so on:

$$\begin{pmatrix} \frac{1}{6} & 0 & -\frac{4}{15} & | & 3 \\ -\frac{1}{27} & \frac{1}{9} & \frac{8}{135} & | & 2 \\ \frac{1}{27} & -\frac{1}{9} & \frac{19}{135} & | & \checkmark{} \end{pmatrix} \rightarrow{} \begin{pmatrix} \frac{1}{6} & 0 & -\frac{4}{15} & | & 3 \\ -\frac{1}{27} & \frac{1}{9} & \frac{8}{135} & | & \checkmark{} \\ \frac{1}{27} & -\frac{1}{9} & \frac{19}{135} & | & \checkmark{} \end{pmatrix} \rightarrow{} \begin{pmatrix} -\frac{4}{15} & 0 & \frac{1}{6} & | & \checkmark{} \\ \frac{8}{135} & \frac{1}{9} & -\frac{1}{27} & | & \checkmark{} \\ \frac{19}{135} & -\frac{1}{9} & \frac{1}{27} & | & \checkmark{} \end{pmatrix}$$

The same inverse matrix WolframAlpha gave us before!

# Why is this algorithm good?

The key reason that this PLU inversion algorithm is preferable over other algorithms is the low number of divisions that is involved, combined with the partial pivoting. In the entire algorithm above, we need to divide by only three different numbers: the diagonal elements in the upper triangular matrix $U$. Divisions are always a bit problematic for numerical algorithms, as their round off error is theoretically unbound, unlike the round off error for additions, subtractions and multiplications that is guaranteed to stay within some limits. Every division could hence lead to a serious loss of accuracy, especially if the numbers in the denominator of the division are small.

In a naive matrix inversion algorithm, the number of divisions might be lower, but the denominators of the divisions are fixed at the start and could hence be anything. In the PLU inversion algorithm we have some control over the denominators of the divisions, as the elements on the diagonal of $U$ will depend on the permutations we perform during the PLU decomposition. In other words: we can choose which numbers we want to use for divisions when choosing the pivot for a row. If we choose the pivot wisely (the element in the column with the highest absolute value like we did), then we can significantly reduce the round off error that occurs during the later inversion of the $U$ matrix. If this still leads to significant round off error, then this can only be because the matrix elements are very small, and numerical inversion would be tricky with any algorithm.

As already mentioned, a second advantage of the PLU inversion algorithm is that it requires very little memory: almost all intermediate steps can simply be stored in the matrix itself, and we only require two additional arrays with the length of the 1D size of the matrix: the pivot array and the temporary array required to store new columns during the third step of the algorithm. This is especially important for large matrices.

Professional astronomer.