In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems...
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In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform. They were...
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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...
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an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix S writes as exponential of a skew-symmetric block matrix Σ {\displaystyle...
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to exponentiation, including: Exponential function, also: Matrix exponential, the matrix analogue to the above Exponential decay, decrease at a rate proportional...
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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...
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Geometric series (section Matrix exponential)
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...
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rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation. While a rotation matrix is an orthogonal matrix R T...
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determine the product of exponentials for a kinematic chain, with the goal of parameterizing an affine transformation matrix between the base and tool...
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Trace (linear algebra) (redirect from Exponential trace)
eigenvalues), one can derive a relation between the trace function, the matrix exponential function, and the determinant: det ( exp ( A ) ) = exp ( tr ...
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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...
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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...
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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...
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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...
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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...
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orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal...
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can be used to compute the matrix exponential eA, a need frequently arising in solving linear differential equations, matrix logarithms and square roots...
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Lie group (redirect from Matrix Lie group)
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...
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\end{aligned}}} This approach can be generalized to matrix exponential and other matrix functions that can be defined as power series. For example...
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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...
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Pauli matrices (redirect from Pauli matrix)
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...
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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...
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Series (mathematics) (section Matrix exponential)
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)...
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sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential type Exponentially equivalent...
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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...
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Rodrigues' rotation formula (section Matrix notation)
s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} ). In terms of the matrix exponential, R = exp ( θ K ) . {\displaystyle \mathbf {R} =\exp(\theta \mathbf...
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Furthermore, exp A {\displaystyle \exp {\mathbf {A} }} is the matrix exponential. Spectral matrices are matrices that possess distinct eigenvalues...
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satisfy ATJA = J. Thus, the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplectic matrix is not necessarily Hamiltonian...
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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...
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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...
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