Efficiently Compute the Inverse of a Matrix Using Tricky Techniques

In summary, it is being discussed whether the inverse of a matrix C can be computed directly from the inverse of matrix A without going through the inverse operation twice. It is mentioned that C is not necessarily invertible, so the answer to this question is "no". However, if A is positive definite, then B and C should also be positive definite. It is suggested that the inverse of C can potentially be computed directly from A by using determinants and submatrices, but it is unlikely to find a general result for this.
  • #1
peterlam
16
0
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!
 
Mathematics news on Phys.org
  • #2
C is not necessarily invertible, so the answer to your question is "no".

For example
[tex]B = \begin{matrix} 1 & -1 \cr 1 & 1 \end{matrix}[/tex]

[tex]C = \begin{matrix} 1 & 1 \cr 1 & 1 \end{matrix}[/tex]
 
  • #3
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!
 
  • #4
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.
 
  • #5


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.
 

What is a tricky matrix inverse?

A tricky matrix inverse refers to the process of finding the inverse of a matrix that is not easily invertible. This can happen due to the matrix being singular or having complex numbers.

Why is finding the inverse of a matrix important?

Finding the inverse of a matrix is important because it allows us to solve systems of linear equations, compute determinants, and perform other mathematical operations that would be difficult without the inverse.

What are some common strategies for solving a tricky matrix inverse?

Some common strategies for solving a tricky matrix inverse include using Gaussian elimination, finding the adjugate matrix, and using eigenvalues and eigenvectors.

Can all matrices be inverted?

No, not all matrices can be inverted. A matrix must be square and have a nonzero determinant in order to have an inverse. If the determinant is zero, the matrix is singular and cannot be inverted.

How can I check if my matrix inverse is correct?

You can check if your matrix inverse is correct by multiplying the original matrix by the inverse. The resulting matrix should be the identity matrix. Another way to check is by calculating the determinant of the inverse, which should be the reciprocal of the determinant of the original matrix.

Similar threads

  • General Math
Replies
13
Views
1K
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
2
Views
984
  • General Math
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
512
  • Calculus and Beyond Homework Help
Replies
1
Views
613
  • Math POTW for Graduate Students
Replies
1
Views
408
  • Precalculus Mathematics Homework Help
2
Replies
57
Views
3K
  • Programming and Computer Science
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
8
Views
993
Back
Top