Therefore, we need only know how to compute the matrix exponential of a Jordan block.

The matrix exponential has applications to systems of linear differential equations.

The matrix exponential is none of these.

The matrix exponential is not surjective when seen as a map from the space of all nxn matrices to itself.

Note that the exponentials are matrix exponentials.

One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations.

For Hermitian matrices there are two notable theorems related to the trace of matrix exponentials:

In addition to providing a computational tool, this formula demonstrates that a matrix exponential is always an invertible matrix.

The Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix.

The matrix exponential of a skew-symmetric matrix A is then an orthogonal matrix R:

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

Under this definition the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.

Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal).

For matrix Lie groups, the elements of g and G are square matrices and the exponential map is given by the matrix exponential.

Finding reliable and accurate methods to compute the matrix exponential is difficult, and this is still a topic of considerable current research in mathematics and numerical analysis.

In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix.

A matrix B is a logarithm of a given matrix A if the matrix exponential of B is A:

J.E. Cohen, S. Friedland, T. Kato, F. Kelly, Eigenvalue inequalities for products of matrix exponentials, Linear algebra and its applications, Vol.

Note that if X is a 1x1 matrix the matrix exponential of X is a 1x1 matrix consisting of the ordinary exponential of the single element of X.

As the nxn generalised shift matrices we are using become zero when raised to power n, when calculating the matrix exponential we need only consider the first n + 1 terms of the infinite series to obtain an exact result.

Due to its simple construction by factorials, a very basic representation of Pascal's triangle in terms of the matrix exponential can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, .

The strongly continuous semigroup T with generator A is often denoted by the symbol e. This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via functional calculus (for example, via the spectral theorem).