A quaternionic matrix is a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

whose elements are

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

Matrix operations

The quaternions form a

noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

, and therefore

addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...

and

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...

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: :

(A+B)_=A_+B_.\,

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 ''i''th row and ''j''th column of the product is the

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...

of the ''i''th row of the first matrix with the ''j''th column of the second matrix. Specifically: :

(AB)_=\sum_s A_B_.\,

For example, for :

U =
\begin
  u_ & u_\\
  u_ & u_\\
\end,
\quad
V =
\begin 
  v_ & v_\\
  v_ & v_\\
\end,

the product is :

UV =
\begin
  u_v_+u_v_ & u_v_+u_v_\\
  u_v_+u_v_ & u_v_+u_v_\\
\end.

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 may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...

for this multiplication is, as expected, the diagonal matrix I = diag(1, 1, ... , 1). Multiplication follows the usual laws of

associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...

and

distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...

. The trace of a matrix is defined as the sum of the diagonal elements, but in general :

\operatorname(AB)\ne\operatorname(BA).

Left scalar multiplication, and right scalar multiplication are defined by :

(cA)_=cA_, \qquad (Ac)_=A_c.\,

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

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

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 :

\begin~~a+bi & c+di \\ -c+di & a-bi \end.

This defines a map Ψ_''mn'' from the ''m'' by ''n'' quaternionic matrices to the 2''m'' by 2''n'' 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 if and only if its determinant is nonzero.

Applications

Quaternionic matrices are used in

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

and in the treatment of multibody problems.

References

{{Matrix classes Matrices Linear algebra