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

In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform. They were...

an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix S writes as exponential of a skew-symmetric block matrix Σ {\displaystyle...

is the identity matrix. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra...

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

rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation. While a rotation matrix is an orthogonal matrix R T...

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...

to exponentiation, including: Exponential function, also: Matrix exponential, the matrix analogue to the above Exponential decay, decrease at a rate proportional...

exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie group, the exponential map reduces to the matrix exponential...

eigenvalues), one can derive a relation between the trace function, the matrix exponential function, and the determinant: det ( exp ⁡ ( A ) ) = exp ⁡ ( tr ⁡...

written as U = eiH, where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix. For any nonnegative integer n, the set...

formal matrix exponential solution may be reduced to a simple form. Below, this solution is displayed in terms of Putzer's algorithm. The matrix equation...

\end{aligned}}} This approach can be generalized to matrix exponential and other matrix functions that can be defined as power series. For example...

determine the product of exponentials for a kinematic chain, with the goal of parameterizing an affine transformation matrix between the base and tool...

can be used to compute the matrix exponential eA, a need frequently arising in solving linear differential equations, matrix logarithms and square roots...

squaring method for the matrix exponential revisited. SIAM review, 51(4), 747-764. How and How Not to Compute the Exponential of a Matrix Nicolas Bourbaki (1989)...

Lie group is the exponential map, which is defined using the standard matrix exponential series for eA For any skew-symmetric matrix A, exp(A) is always...

orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal...

\mathbf {y} } . If A {\displaystyle A} is skew-Hermitian, then the matrix exponential e A {\displaystyle e^{A}} is unitary. The space of skew-Hermitian...

satisfy ATJA = J. Thus, the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplectic matrix is not necessarily Hamiltonian...

sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential type Exponentially equivalent...

then the exponential map takes the Lie algebra of G {\displaystyle G} into ⁠ G {\displaystyle G} ⁠; thus, we have an exponential map for all matrix groups...

defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a...

idempotent. The matrix exponential of J is exp ⁡ ( μ J ) = I + e μ n − 1 n J {\displaystyle \exp(\mu J)=I+{\frac {e^{\mu n}-1}{n}}J} The all-ones matrix arises...

x and y are replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for...

the exponential itself is just 1, which makes it the generic group element of SU(2). A more abstract version of formula (2) for a general 2 × 2 matrix can...

trickier due to the matrix exponential integral. It can, however, be computed by first constructing a matrix, and computing the exponential of it F = [ − A...

mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. The exponential of a variable...

OEIS). A Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix. The example below...

square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the...