Moore Determinant Of A Quaternionic Hermitian Matrix
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In mathematics, the Moore determinant is a
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
defined for
Hermitian matrices In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
over a
quaternion algebra In mathematics, a quaternion algebra over a field (mathematics), field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension (vector space), dimension 4 ove ...
, introduced by . Because quaterion multiplication does not commute, it is necessary to specify the order in which multiplication occurs. The Moore determinant uses the formal classical determinant, which has n! terms consisting of products of elements of the matrix, and for each term specifies an order for those elements to be multiplied. Specifically, it separates out cycles of factors a_,a_,\dots,a_. The shortest cycles are placed first, with the smallest index within the cycle occurring first. Ties in the length of the cycle are broken by listing the cycle with the smallest f_1 first. This definition has the property that the Moore determinant of a matrix formed from a suitable collection of vectors of quaternions is zero if and only if the vectors are linearly dependent.


See also

*
Dieudonné determinant In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by . If ''K'' is a division ring, then the Dieudonné determinant is a group homomor ...
*
quasideterminant In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows: : \left, \begin a_ & a_ \\ a_ & a_ \end \_ = a_ - ...


References

* Matrices (mathematics) {{matrix-stub