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A nilpotent matrix A is a square matrix such that for some positive integer p. It is said to be of index p if p the least positive integer for which . For example, the matrix:

is nilpotent with index 2. More generally, a triangular matrix of size n with zeros along the main diagonal is nilpotent with index . For example, the follow matrix is nilpotent with index 3:

The determinant and trace of a nilpotent matrix are always zero. Thus, such matrices are not invertible. However, and are invertible where N is a nilpotent matrix:

(5.16)

where I is the identity matrix. Since there are only finitely many non-zero terms, we see that both sums converge.

An idempotent matrix A is one for which . Examples are:

Idempotent matrices arise in regression analysis and econometrics, for example in ordinary least squares problems, in particular when estimating sums of squared residuals.

An involutory matrix is one that is its own inverse:

An example is:

For example, the Pauli matrices are involutory: