Efficiently Compute the Inverse of a Matrix Using Tricky Techniques

[peterlam]

Suppose A is a invertible n-by-n matrix. Let B be the inverse of A, i.e. B = A^(-1).It is trivial that A = B^(-1).

If we construct a matrix C whose entry is the square of corresponding entry of B, i.e. C_ij = (B_ij)^2, then we compute the inverse of C.

We can compute the inverse of C directly from A without going through the inverse operation twice?

Thank you!

 

[AlephZero]

C is not necessarily invertible, so the answer to your question is "no".

For example
$$B = \begin{matrix} 1 & -1 \cr 1 & 1 \end{matrix}$$

$$C = \begin{matrix} 1 & 1 \cr 1 & 1 \end{matrix}$$

 

[peterlam]

What if we only consider A is positive definite? Then B is positive definite and C should be positive definite too.

Can we compute the inverse of C directly from A in this case?

Thank you!

 

[AlephZero]

In my counterexample B is positive definite.

x^T B x = x_1^2 + x_2^2

You can write any inverse matrix explicitly in terms of determinants of the matrix and submatrices (this is equivalent to Cramer's rule for solving equations). Think about how a derminant is calculated, and what happens to it if you square all the entries in the matrix. I think it is very unlikely you will get any general result about this.

 

[CellCoree]

I am happy to see that you are exploring efficient techniques for computing the inverse of a matrix. This is an important problem in many areas of science and engineering, and finding efficient solutions can greatly improve computational efficiency and accuracy.

To address your content, yes, it is true that A = B^(-1) and we can compute the inverse of C directly from A without going through the inverse operation twice. This is because the inverse of a matrix can be calculated by using the adjugate matrix, which is constructed using the transpose of the matrix of cofactors. In this case, the matrix of cofactors for B would also be the matrix of cofactors for C, since the entries of B and C are squared versions of each other.

This approach can be seen as a type of shortcut or trick, as it avoids the need to calculate the inverse twice. However, it is important to note that this method may not always be the most efficient or accurate solution for computing the inverse. It is always important to carefully consider the properties and structure of the matrix in question before choosing a specific method for computing its inverse.

In conclusion, the approach you have described can be a useful technique for efficiently computing the inverse of a matrix, but it may not always be the most appropriate solution. As scientists, it is important for us to continually explore and evaluate different techniques and methods to find the most efficient and accurate solutions for our problems.