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→Relationship to skew-symmetric matrices: Added 3D relationships with Euler-Rodrigues and quaternions Tags: Mobile edit Mobile web edit |
→Relationship to skew-symmetric matrices: clearer connection with quaternions Tags: Mobile edit Mobile web edit |
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which corresponds exactly to the polar form <math>\cos \theta + i \sin \theta = e^{i \theta}</math> of a complex number of unit modulus.
In 3 dimensions, the matrix exponential is [[Rodrigues' rotation formula#Matrix notation|Rodrigues' rotation formula in matrix notation]], and when expressed via the [[Euler-Rodrigues formula]], the algebra of its four parameters
The exponential representation of an orthogonal matrix of order <math>n</math> can also be obtained starting from the fact that in dimension <math>n</math> any special orthogonal matrix <math>R</math> can be written as <math>R = QSQ^\textsf{T},</math> where <math>Q</math> is orthogonal and S is a [[block matrix#Block diagonal matrix|block diagonal matrix]] with <math display="inline">\lfloor n/2\rfloor</math> blocks of order 2, plus one of order 1 if <math>n</math> is odd; since each single block of order 2 is also an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix ''S'' writes as exponential of a skew-symmetric block matrix <math>\Sigma</math> of the form above, <math>S = \exp(\Sigma),</math> so that <math>R = Q\exp(\Sigma)Q^\textsf{T} = \exp(Q\Sigma Q^\textsf{T}),</math> exponential of the skew-symmetric matrix <math>Q\Sigma Q^\textsf{T}.</math> Conversely, the surjectivity of the exponential map, together with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices.
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