An idempotent matrix is a square matrix that remains unchanged when multiplied by itself. This makes it a type of singular matrix, apart from the identity matrix.

Idempotent Matrixes

Idempotent matrices are useful for some mathematical computations, especially in the fields of linear statistical models and econometrics. They also play a role in the construction of various least squares estimators and in the analysis of variance.

A matrix is a definite collection of numbers, symbols, or expressions arranged in a tabular form with rows and columns. This is done so that it can be used for a particular application or to represent a specific mathematical object.

Idempotent Matrix Example

The idempotency of a matrix plays a crucial role in the development of various linear models and in econometrics, where it is essential for certain calculations such as the estimation of the [M] matrix and the Hat matrix during regression analysis. Hence, it is necessary to know about these matrices in order to properly use them.

For instance, one can calculate the determinant of an idempotent matrix using the l-by-l formula. This is important for evaluating the variance of a regression estimator.

A 2 x 2 displaystyle 2times2 matrix is idempotent if and only if it has a diagonal trace and all of its non-diagonal elements are either zero or 1. The trace is a value equal to the rank of the matrix.

Matrix Squared Equals Itself

Moreover, the eigenvalues of an idempotent matrix are 0 or 1 depending on its non-diagonal elements. If a non-diagonal element is nonzero, then its eigenvalue is 0; if the element is zero, then its eigenvalue is 1.

Some idempotent matrices are also square. This means that they can be divided into parts by a given number of rows and columns, or that they have a bounded number of entries. The idempotency of these matrices is also conserved in the case of a change of basis.

These idempotent matrices have an important relationship with nilpotent matrices. This is a topic that I haven’t covered in depth, but can be found here:

In this chapter, I present an elementary characterization of the idempotency of triangular matrices with entries from an arbitrary ring and demonstrate some structure that can lead to the explicit enumeration or implicit counting of all such idempotent matrices. The result reveals some basic properties of idempotent matrices and provides a convenient method for constructing such matrices.

The main result of the paper is that for every field F and for each pair (n,k) of positive integers, there is a singular nxn matrix S over F which is a product of k idempotent matrices over F iff rank(I – S)k* nullity S. This is a fundamental result in the theory of linear regression analysis and in econometrics, and it may have applications beyond the field of mathematics.

Amy Shelton