Compute the core-nilpotent decomposition of a square matrix:

Format the result:

Core-nilpotent decomposition of a 4×4 matrix:

Verify the relation with m:

Solve the matrix differential equation , with singular coefficients:

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

Compute the core-nilpotent decomposition of the solution to :

Let :

The solution is then , where is the solution to :

Compare with the result given by DSolveValue:

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

The matrix is singular:

Using the core-nilpotent decomposition , let :

The solution is , where is an arbitrary vector:

Verify the solution:

CoreNilpotentDecomposition returns a triple {t,c,n}:

The matrix c is nonsingular:

The matrix n is nilpotent:

The original matrix m can be expressed in terms of its core-nilpotent decomposition:

The core part of the decomposition for an invertible matrix is equal to the matrix:

The nilpotent part of the decomposition is an empty list:

The similarity matrix t is taken to be the identity matrix:

The identity expressed using BlockDiagonalMatrix holds nonetheless:

The nilpotent part of the decomposition for a nilpotent matrix is equal to the matrix:

The core part of the decomposition is an empty list:

The similarity matrix t is taken to be the identity matrix:

The identity expressed using BlockDiagonalMatrix holds nonetheless:

DrazinInverse can be computed with CoreNilpotentDecomposition:

Verify the equality :

The core-nilpotent decomposition is not unique:

Either or , but not both, can be equal to {}:

MatrixQ[{}] gives False:

Use BlockDiagonalMatrix to reconstruct the original matrix, since it interprets {} as a 0×0 matrix: