Quaternionic matrix
Encyclopedia
A quaternionic matrix is a matrix
whose elements are quaternion
s.
and multiplication
can be defined for quaternionic matrices as for matrices over any ring.
Addition. The sum of two quaternionic matrices A and B is defined in the usual way by element-wise addition:
Multiplication. The product of two quaternionic matrices A and B also follows the usual definition for matrix multiplication. For it to be defined, the number of columns of A must equal the number of rows of B. Then the entry in the ith row and jth column of the product is the dot product
of the ith row of the first matrix with the jth column of the second matrix. Specifically:
For example, for
the product is
Since quaternionic multiplication is noncommutative, care must be taken to preserve the order of the factors when computing the product of matrices.
The identity
for this multiplication is, as expected, the diagonal matrix I = diag(1, 1, ... , 1). Multiplication follows the usual laws of associativity
and distributivity
. The trace of a matrix is defined as the sum of the diagonal elements, but in general
Left scalar multiplication is defined by
Again, since multiplication is not commutative some care must be taken in the order of the factors.
for (square) quaternionic matrices so that the values of the determinant are quaternions. Complex valued determinants can be defined however. The quaternion a + bi + cj + dk can be represented as the 2×2 complex matrix
This defines a map Ψmn from the m by n quaternionic matrices to the 2m by 2n complex matrices by replacing each entry in the quaternionic matrix by its 2 by 2 complex representation. The complex valued determinant of a square quaternionic matrix A is then defined as det(Ψ(A)). Many of the usual laws for determinants hold; in particular, an n by n matrix is invertible exactly when its determinant is nonzero.
and in the treatment of multibody problems.
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
whose elements are quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s.
Matrix operations
The quaternions form a noncommutative ring, and therefore additionMatrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum....
and multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
can be defined for quaternionic matrices as for matrices over any ring.
Addition. The sum of two quaternionic matrices A and B is defined in the usual way by element-wise addition:
Multiplication. The product of two quaternionic matrices A and B also follows the usual definition for matrix multiplication. For it to be defined, the number of columns of A must equal the number of rows of B. Then the entry in the ith row and jth column of the product is the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of the ith row of the first matrix with the jth column of the second matrix. Specifically:
For example, for
the product is
Since quaternionic multiplication is noncommutative, care must be taken to preserve the order of the factors when computing the product of matrices.
The identity
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
for this multiplication is, as expected, the diagonal matrix I = diag(1, 1, ... , 1). Multiplication follows the usual laws of associativity
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
and distributivity
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
. The trace of a matrix is defined as the sum of the diagonal elements, but in general
Left scalar multiplication is defined by
Again, since multiplication is not commutative some care must be taken in the order of the factors.
Determinants
There is no natural way to define a determinantDeterminant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
for (square) quaternionic matrices so that the values of the determinant are quaternions. Complex valued determinants can be defined however. The quaternion a + bi + cj + dk can be represented as the 2×2 complex matrix
This defines a map Ψmn from the m by n quaternionic matrices to the 2m by 2n complex matrices by replacing each entry in the quaternionic matrix by its 2 by 2 complex representation. The complex valued determinant of a square quaternionic matrix A is then defined as det(Ψ(A)). Many of the usual laws for determinants hold; in particular, an n by n matrix is invertible exactly when its determinant is nonzero.
Applications
Quaternionic matrices are used in quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and in the treatment of multibody problems.