# How to formulate nonsingularity of matrix (I + A*B) in LMIs?

• Amir Hossein
In summary, a nonsingular matrix is a square matrix with a unique inverse and is important in control theory and optimization problems. It can be checked by calculating its determinant or rank, and its formulation in LMIs is (I + A*B) > 0. This can be done for any values of A and B, but feasibility of the solution is dependent on the values of A and B.
Amir Hossein
Consider I+A*B where A: (n*l) is a variable matrix and B: (l*n) is known. I am looking for some way to find a sufficient condition for nonsingularity of I+A*B

## 1. What is a nonsingular matrix?

A nonsingular matrix is a square matrix that has a unique inverse. This means that the matrix can be multiplied with its inverse to give the identity matrix (a matrix with 1s on the diagonal and 0s elsewhere).

## 2. What is the importance of formulating nonsingularity of matrix (I + A*B) in LMIs?

Formulating nonsingularity of matrix (I + A*B) in LMIs (Linear Matrix Inequalities) is important in control theory and optimization problems. It allows for the use of convex optimization techniques to find solutions, which are more efficient and reliable compared to traditional methods.

## 3. How do I check if a matrix is nonsingular?

A matrix can be checked for nonsingularity by calculating its determinant. If the determinant is equal to 0, then the matrix is singular. Alternatively, the rank of the matrix can also be used to determine its nonsingularity - a matrix with full rank is nonsingular.

## 4. What is the formula for formulating nonsingularity of matrix (I + A*B) in LMIs?

The formula for formulating nonsingularity of matrix (I + A*B) in LMIs is: (I + A*B) > 0, where > 0 indicates that the matrix is positive definite.

## 5. Can nonsingularity of matrix (I + A*B) be formulated for any values of A and B?

Yes, nonsingularity of matrix (I + A*B) can be formulated for any values of A and B as long as the dimensions of the matrices are compatible for multiplication and the matrices are square. However, the formulation may not always have a feasible solution, depending on the values of A and B.

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