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Complex and Hypercomplex Numbers
Real-element Matrix Representation

Table of Contents

1. The Most Elementary Group
2. Complex Numbers in Real valued Matrix Representation
3. Hypercomplex Numbers

1. The Most Elementary Group

Quaternions", "hypernumbers" and "hypercomplex numbers" are historical predecessors to vector analysis (invented by William Hamilton in mid seventieth century). There is a deep mathematical connection between these, the linear algebra of matrices, and the group theory. A three-dimensional application of such an objects is the rotation operator, as it is expressed in the form of a vector (a "pseudovector") of complex-element matrices, called "pseudoscalars". This group has the same properties as the older concept called "hypercomplex numbers", and thus is a matrix representation for them. The most often referenced representation for them is a complex-element matrix group, better known as the "Pauli spin matrices".

What are the most elementary objects in mathematics? One could be tempted to say that they are the numbers. But numbers are not that elementary really. They have complicated relations to each other. Operations with a set of numbers tend to give results in numbers that are outside of the given set, and so on. But there is more elementary mathematical objects, called "groups". A group is the mathematical notion of objects that are characterized by operations, "additions" and "multiplications", where the results of the operations keep inside the same set of objects. Which is then the most elementary of all groups?

A group is defined by the operations alone, by the "multiplication table". When the elements are given some kind of individual appearance, that may be matrices or rotation operators or anything, it is called a "representation" of the group. The most elementary group that is not trivial, that has interesting properties to investigate, is a group of three base elements. One such a group is the three-dimensional rotation group, that was the first representation known historically. It is a group of rotation operators that are able to describe all rotations in a three-dimensional space. But the group itself is also known with a modern group-theoretical name as SU(2), the two-parameter special unitary group.

I'd like to introduce here a real-element matrix representation for the most elementary nontrivial group. Why real-element? Because these real-element matrices actually can act as solutions to equations that traditionally call for complex numbers. And as a consequence, complex numbers too can be given a real-element matrix representation. And we see that the complex numbers themselves in fact are members of the SU(2) group, which is not so obvious otherwise.

Consider the set of unitary 2×2 matrices of real valued elements. The set is a complete set of base members, meaning that none of the base matrices can be reproduced from the others with any linear combination, and any other real-element 2×2 matrices can be reproduced as a linear combination of these four matrices. (Note the obvious similarity of appearance compared with the Pauli spin matrices; in fact only I₂ is different.):

Determinants are equal to +1 or -1 , which determines what is called unitarity.

Matrix multiplication gives the multiplication table, which, as we see, is very similar (actually, as identical as it can be without the imaginary unit explicitly expressible) witch that of Pauli spin matrices.

Multiplication of two of the members of the group gives a third member of the group, or its negative. There is a negative member for every member that also belongs to the group, according to the group theory. (There is also a zero-member in the group, in our case a zero matrix.) Note that the multiplication is not commutative, changing the order in multiplication changes the sign of the result.

Note: Hypercomplex base vectors in the traditional form have somewhat different table:

and the nearly related Hamilton's quaternions have the multiplication table:

Let us introduce now a conjugate product, a product of a matrix and a transposed one:

All of the matrices I_n has their conjugate product with themselves equal to the unit matrix I₀:

2. Complex Numbers in Real valued Matrix Representation

We can treat the matrices as a kind of "numbers", mathematically often called "pseudoscalars". They can formally be given the role to represent numbers. Now, because multiplication with I₀ does not change any matrix, I₀ can be given the role to represent the number 1. (The zero matrix represents the number zero.) Note that all of the matrices, I₀ , I₁ , and I₃ satisfy the equation:

But I₂ alone represent a solution to the equation:

On the other hand, this solution to (2.2) is known to be , that is, the imaginary unit  i . So the matrix I₂ can be taken to be a matrix representation for the imaginary unit. Then any complex number can be expressed with the two matrices I₀ and I₂.
Taking a complex number of form z = a + bi , it can now be represented by a matrix:

All of the elements of this matrix are real valued. It is the matrix itself, not its components, that is taken to be the complex number. So the complex number z is represented by a real element "pseudoscalar" matrix. The absolute value of z is the the square root of the determinant of the matrix:

If  z  is a solution to the basic algebraic equations above it is unimodular, its determinant is
 +1 or -1 . These equations (2.1) and (2.2) are basic in the meaning that if we have the solutions to these simple equations, we can construct from them all the solutions to all of the algebraic equations. That was the reason why imaginary numbers originally were invented. Now we use here these pseudonumbers in stead.

We get the complex conjugate of  z  by changing the sign of the imaginary part, which now means changing the sign of the non-diagonal elements, which represent the imaginary part. This on the other hand is identical to the transpose of the matrix (switching columns to rows and vice versa).

which corresponds to the complex conjugate of (a + bi) which is (a - bi).

Note that the complex matrix representation automatically Hermitian (self adjoint). The Hermitian adjoint is defined as the transpose of the complex conjugate of a matrix, which in our case this is the same as transposing twice, and thus gives the original matrix. The condition for a matrix to be Hermitian is that its adjoint is equal to the matrix self. This is then trivially true for the pseudoscalar complex representation..

We get the absolute value of a complex number z by multiplicating of the complex matrix with its transpose (conjugate product).

The inverse of z becomes

Separation of the Real and the Imaginary part:

Conversion from pseudoscalars to real numbers:

The argument q of a complex pseudoscalar number z is given by the identities:

By addition and subtraction, and using results (2.11) and (2.12) above about separating of the real and imaginary part:

Then by division we get an expression for the argument angle:

3. Hypercomplex Numbers

The complex matrix representation presented in chaper 2. does not include all of the 2×2 base matrices. More generally let us define a hypercomplex number:

We now have here a complete linear system (manifold) of pseudonumbers, in fact constituting of a vector (pseudovector). In matrix form any of them can be written as a linear combination of a set of basis pseudonumber matrices:

The determinant of this matrix becomes:

Note that the determinant may in certain cases be zero even for nonzero matrices. That is when the matrix is singular, rising a need of special conditions that the matrix elements must fulfil to avoid singular matrices. This is commonplace matrix algebra and we don't need now to go deeper in that problem. Assuming the matrix is non-singular, the inverse of Z becomes:

Comparison with the original matrix reveals that the inverse can be done directly in either of the two ways:

1.a changing the sign before the basis matrices except the first one that is the unit matrix;

or , which is the same as

1.b swapping the diagonal elements and swapping and changing sign of the off-diagonal element;

and then finally,

2. multiplying the resulting matrix with the inverse value of the determinant of the original matrix.

Complex conjugation is related to the inverse. For the commonplace complex numbers a + b i the complex conjugation a - b i can be written as

One good reason for defining the concept complex conjugation is that it is needed to produce the norm of a complex number:

In a similar manner we define the norm of a hypercomplex number:

where the hypercomplex conjugation is defined as

Then substituting the inversion from the formula (3.4) , the hypercomplex conjugation becomes

For forming the hypercomplex conjugation the same rules apply as above for the inverse, except for the multiplying factor, let us call it a nominal function, which is now

The matrix transpose of a hypercomplex number become

Transposing of a matrix does not affect the determinant, neither the norm, so

Performing both transpose and conjugation we now call the adjoint of the hypercomplex number. Here is where the hypercomplex notion differs from the matrix notion.The matrix is a representation of the former, but the commonplace matrix adjoint does not apply here. We can not use the commonplace matrix conjugation (which would be equal to the original matrix because its elements are real). In stead we have the hypercomplex conjugation defined above.

The hypercomplex numbers are a kind of pseudonumbers that have their corresponding inverse, conjugation, and adjoint, that also are hypercomplex numbers. Transpose is a matrix property, not really a hypercomplex property. The norm is a hypercomplex property, whereas determinant is a matrix property. The nominal function gives the conversion rule between these two.

Using the transpose above and the conjugation rule we get for the hypercomplex adjoint:

The other way round, starting from the conjugate and transposing it we get

So we get the same matrix in either way, and denote this result the hypercomplex adjoint:

When the adjoint is identical to the inverse of a hypercomplex number, then the hypercomplex number is said to be unitary .

Succestions for "normalized" forms in expressing the hypercomplex matrix:

Write the hypercomplex number Z in the form using the unimodular matrix X .
Assuming that the matrix representation of Z is non-singular, and denoting: