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  2. Vectorization (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Vectorization_(mathematics)

    For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal.

  3. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...

  4. Transformation matrix - Wikipedia

    en.wikipedia.org/wiki/Transformation_matrix

    In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...

  5. Change of basis - Wikipedia

    en.wikipedia.org/wiki/Change_of_basis

    Normally, a matrix represents a linear map, and the product of a matrix and a column vector represents the function application of the corresponding linear map to the vector whose coordinates form the column vector. The change-of-basis formula is a specific case of this general principle, although this is not immediately clear from its ...

  6. Rotation formalisms in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_formalisms_in...

    The Euler axis can be also found using singular value decomposition since it is the normalized vector spanning the null-space of the matrix I − A. To convert the other way the rotation matrix corresponding to an Euler axis ê and angle θ can be computed according to Rodrigues' rotation formula (with appropriate modification) as follows

  7. Rodrigues' rotation formula - Wikipedia

    en.wikipedia.org/wiki/Rodrigues'_rotation_formula

    Vector geometry of Rodrigues' rotation formula, as well as the decomposition into parallel and perpendicular components. Let k be a unit vector defining a rotation axis, and let v be any vector to rotate about k by angle θ ( right hand rule , anticlockwise in the figure), producing the rotated vector v rot {\displaystyle \mathbb {v} _{\text ...

  8. Commutation matrix - Wikipedia

    en.wikipedia.org/wiki/Commutation_matrix

    In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K (m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(A T): K (m,n) vec(A) = vec(A T) .

  9. Whitening transformation - Wikipedia

    en.wikipedia.org/wiki/Whitening_transformation

    A whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are uncorrelated and each have variance 1. [1]