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...

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...

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

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

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...

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

mathematical tools such as the Taylor series, the Fourier series, and the matrix exponential. The geometric series a + a r + a r 2 + a r 3 + … {\displaystyle...

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

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

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...

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...

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...

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...

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...

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

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

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

The exponential function is a mathematical function denoted by f ( x ) = exp ⁡ ( x ) {\displaystyle f(x)=\exp(x)} or e x {\displaystyle e^{x}} (where...

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...

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...

when inverting the exponential map, that is, when finding a rotation vector that corresponds to a given rotation matrix. The exponential map is onto but...

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)...

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

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...

s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} ). In terms of the matrix exponential, R = exp ⁡ ( θ K ) . {\displaystyle \mathbf {R} =\exp(\theta \mathbf...

Furthermore, exp ⁡ A {\displaystyle \exp {\mathbf {A} }} is the matrix exponential. Spectral matrices are matrices that possess distinct eigenvalues...

unit matrix, showing that the rapidity is the coordinate on this Lie algebra. This action may be depicted in a spacetime diagram. In matrix exponential notation...

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

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...