It is relatively easier to find a solution to a system of linear equations in the form of given the matrix . But what systematic ways are there that allows us to obtain a matrix given a equation?
For example, consider the following equations with all terms existing in
Although it is easy to see that with all other terms being 0 is a viable solution, I am curious if there is a more systematic way of finding a matrix that satisfies a equation. Even more importantly, how should these methods be adapted when there are added constraints on the properties of the matrix? For example, if we require that the matrix of interest should be invertible, or of rank = k?
Why I am interested in such question
Consider the vector space , the problem of finding a basis such that can be reduced to a problem that has been stated above.