Introduction to Imaginery Number and Hermitian Matrix
by Kreecha Puphaiboon 30/9/2021
Hi my name is Kreecha Puphaiboon I work for Krungsri Bank in Thailand (https://www.krungsri.com). I am writing this with main objective to introduce:
- Complex
- Hermitian
- and Unitary Matrices
At work, I am working on two projects: 1) clustering for marketing campaings which require optimization for advertising based on time, click through rate and investment cost and 2) voice authentication. The first is multi-dimension clustering in a way. It is great, i have the opportunity to use Complex and Hermitian matrices at the same time for both projects. So, i started talking to many people but not so many undestand, so i though may be i write about it and explain to everyone. It is also nicer to share to the public as a good scientist.
Let’s get started.
1) Complex Matrix
Recall that first complex numbers, Z is a number that can be written as
where a and b are real numbers, called the real and imaginary part of the complex number, and
is called imaginary unit. Imaginary unit is not knowing can be either positive or negative caused by shifting or translating when you dont know the state. For example, when you are in the total darkness in 3D space you dont know you will be moving to left or right and up or down. If we base where you stand on a line number as a starting point the middle based at 0 to the right it is positive and to the left is negative. But when you are in the total darkness you don’t know the direction. So, imaginary numbers are abstract concepts that are used when you need the square root of a negative number.
When we manipulate complex numbers, we often see
People may wonder why we have to deal with -1, let me give you an example. My project dealing with voice, we have to deal with signal which comes in as a sine wave (see below Figure). Let’s say I have a square root form and i want to capture the wave a at every 38 tick (i dont want to process each tick, so i can compute faster), then i randomly pick up a number (x=38) then you can see i pick up -1. This is an example when we sometimes don’t know what we are going to get but you need to square root. In signaling problem, we have to deal with both positive and negative numbers.
Back to Imaginary numbers, they allow us to find solutions to equations that have no real solutions. For example, the equation
has no real solution, but it has two imaginary solutions
To think simply of the fact, a real number can be seen as a complex number with zero imaginary part.
Complex numbers can also be represented in polar form, which has a magnitude term and an angular term. The magnitude of a complex number z=x+iy is given as
, and its angular term is calculated as
Clearly, we can see the above help us to find direction to move as well as the angle. They are quite useful.
Complex conjugate
An important concept is that of complex conjugate. Given a complex number its conjugate, denoted by z̅, is
As a consequence, double conjugation leaves numbers unchanged:
Note that the complex conjugate simply changing the sign of the imaginary part (the real part does not change)
Complex matrices
Complex matrices (and vectors) are matrices whose entries are complex numbers. Complex conjugation of matrices, Given a K × L matrix A, its complex conjugate A̅ is the matrix such that
that is, the (k, l)−th entry of A ̅is equal to the complex conjugate of the (k, l)−th entry of A, for any k ≤ K and l ≤ L. For example, let define the matrix
Then its complex conjugate is
Allow me to repeat if original complex matrix is
, then complex conjugate =
Conjugate transpose or known as Hermitian Transpose
2) Hermitian Matrix
A square matrix is called Hermitian if it is self-adjoint. Hermitian matrices have real eigenvalues whose eigenvectors form a unitary basis. For real matrices, Hermitian is the same as symmetric.
and lastly 3) Unitary Matrix
Possibly, the real use of a unitary matrix is an orthogonal matrix which have significant importance in many areas, sound engineering, engineering and quantum mechanics because they preserve norms, and thus, probability amplitudes. Remember that for real number orthogonal (real) columns form an orthonormal basis
where as unitary (complex) columns form an orthonormal vectors
A real unitary matrix is an orthogonal matrix in fact. What we see if the inverse of a matrix is equal to the identity matrix.
The uses of unitary matrices are beyond the scope of the writing but they are used widely in Spectral and Voice analysis and Quantum mechanic. For a unitary matrix: a) all eigenvalues have absolute value 1. b) eigenvectors corresponding to distinct eigenvalues are orthogonal, c) there is an orthogonal basis of the whole space, consisting of eigenvectors.
I hope the content and code will be helpful to some people.
code: https://github.com/kreecha/medium/tree/master/hermitian
