An example of a semisimple non-unital ring is , the row-finite, column-finite, infinite matrices over a field K.

At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.

Let F be a field, and consider a certain ring of infinite matrices over F.

This method relies on the exact solution of the elastic problem for an inclusion of known geometry (an ellipsoid) surrounded by an infinite matrix.

This idea applied to bounded linear operators on a Banach space, which can be seen as infinite matrices, leads to the holomorphic functional calculus.

Ulm's original proof was based on an extension of the theory of elementary divisors to infinite matrices.

He was suddenly back in his cell within the infinite matrix, but only for a moment.

The infinite symmetric matrix starting:

A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.

A matrix with an infinite number of row or columns (or both) is called an infinite matrix.