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The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
After the algorithm has converged, the singular value decomposition = is recovered as follows: the matrix is the accumulation of Jacobi rotation matrices, the matrix is given by normalising the columns of the transformed matrix , and the singular values are given as the norms of the columns of the transformed matrix .
The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension . [2] [3] The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum. The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices.
the covariance matrix; the set of terminal symbols in a formal grammar; Mathematical surface; Sigma baryon; represents: Stefan–Boltzmann constant in blackbody radiation; the divisor function in number theory; the real part of the complex variable = + in analytic number theory
The matrix ¯ is the Schur complement of Σ 22 in Σ. That is, the equation above is equivalent to inverting the overall covariance matrix, dropping the rows and columns corresponding to the variables being conditioned upon, and inverting back to get the conditional covariance matrix.
A similar canonical set of sigma points can be generated in any number of dimensions by taking the zero vector and the points comprising the rows of the identity matrix, computing the mean of the set of points, subtracting the mean from each point so that the resulting set has a mean of zero, then computing the covariance of the zero-mean set ...
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The smallest singular value of a matrix A is σ n (A). It has the following properties for a non-singular matrix A: The 2-norm of the inverse matrix (A-1) equals the inverse σ n-1 (A). [1]: Thm.3.3 The absolute values of all elements in the inverse matrix (A-1) are at most the inverse σ n-1 (A). [1]: Thm.3.3