Nilpotent Matrices in Jordan Decompositions

In this representation, the greener the square, the larger the entry relative to the others.

A power of a diagonal matrix is the diagonal matrix formed by taking the power of its entries.

Powers of upper- or lower-triangular matrices are again matrices of the same type.

Powers of a tridiagonal matrix spread out from the main diagonal.

A superdiagonal matrix has its nonzero entries above the main diagonal; a subdiagonal matrix has its nonzero entries below. The nonzero entries of powers of either type retreat one diagonal at a time to a corner. Such matrices are nilpotent, meaning that eventually one of their powers is the zero matrix. This shows that matrices are very unlike ordinary numbers.

Thanks to the Jordan decomposition, the calculation of the exponential of a square matrix is very simple. According to this decomposition, the original matrix is equivalent to a matrix with Jordan elementary blocks strung out along the main diagonal, where is real or complex, is an identity matrix, and is a nilpotent matrix with nilpotent index (meaning that ). Nilpotence can be seen by making the powers of "move" the diagonal right and up until it disappears when the power coincides with its order. The decomposition reduces the problem of taking the matrix exponential of a matrix to .