Nilpotent Matrices in Jordan Decompositions

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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 .


Contributed by: Pablo Alberca Bjerregaard (University of Malaga, Spain) and George Beck (March 2011)
Open content licensed under CC BY-NC-SA




Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.