Compute the Drazin inverse of a matrix:

Drazin inverse of a 3×3 matrix:

Verify a few properties of this generalized inverse:

Drazin inverse of a 4×4 matrix:

Verify the definition of DrazinInverse:

Solve the matrix differential equation , with singular coefficients:

Both and are singular, so the equation cannot be put in the standard form :

The solution is , where solves and solves :

Compare with the result given by DSolveValue:

Find the general solution of the matrix difference equation with singular coefficient matrix :

The matrix is singular:

The solution is , where is an arbitrary vector:

Verify the solution:

Compute the effective resistance matrix of a graph:

Compute the effective graph resistance:

DrazinInverse is the same as Inverse for invertible matrices:

DrazinInverse[m] satisfies the relations and :

Verify the required properties:

Unlike PseudoInverse, it is not necessarily the case that :

The other Moore–Penrose equations [more info] need not be satisfied, either:

DrazinInverse is invariant under matrix conjugation, that is :

DrazinInverse can be computed with CoreNilpotentDecomposition:

Verify the equality :

For a diagonal matrix m, DrazinInverse[m] is a diagonal matrix with nonzero elements inverted:

Consider the Jordan matrix given by JordanDecomposition[m]:

DrazinInverse maps blocks with zero diagonals to zero, and other blocks to their inverse:

Moreover, :

Define a function for computing the index of a square matrix:

Compute the index of a matrix:

DrazinInverse[m] satisfies the relation , where k is the index of m:

PseudoInverse[m] can be computed using DrazinInverse as :