The Moore-Penrose pseudoinverse A(+) of a matrix A is a generalization of its inverse A(−1). It's also known as the Moore-Penrose inverse or just the pseudoinverse. When a matrix's determinant is zero, it cannot be inverted. We'd call the matrix singular — it has no inverse. This is unfortunate, because the inverse is valuable in solving a system of equations. However, there is good news — if we can find some value that is almost a solution, we can still do some really useful things. Finding this approximate solution is precisely what the matrix pseudoinverse allows us to do. 1. Click the green flag to initialize the project. 2. Setting Button: setting up the matrix A. 3. Calculate Button: calculating the A(+). 4.Clear Button: clearing the screen.
Project Reference: https://www.omnicalculator.com/math/pseudoinverse How to calculate the pseudoinverse? We can evaluate the pseudoinverse A (+) in many ways. If you use singular value decomposition to obtain the terms of A = U ⋅ S ⋅ V (T), then you can pretty easily calculate A's pseudoinverse with A (+) = V ⋅ S (+) ⋅ U (T). But that's a lot of effort, and so mathematicians have discovered some shortcuts. If A has linearly independent columns, you can calculate the Moore-Penrose pseudoinverse A (+) with A (+) = (A (T) ⋅ A) ^ (−1) ⋅ A (T) Similarly, if A has linearly independent rows, A (+) = A (T) ⋅ (A ⋅ A (T)) ^ (−1) If neither A's columns nor its rows are linearly independent, it gets a little bit trickier! 1. Start by calculating A ⋅ A (T) and row reduce it to reduced row echelon form. 2. Take the non-zero rows of the result and make them the columns of a new matrix P. 3. Similarly, row-reduce A (T) ⋅ A and use its non-zero rows for the columns of the new matrix Q. 4. With your newly found P and Q, calculate M = P (T) ⋅ A ⋅ Q Finally, calculate the pseudoinverse A (+) = Q ⋅ M^ (−1) ⋅ P (T) .
