Complex Conjugate

Every complex number has a complex conjugate, where the real part remains the same, but the imaginary part has the opposite sign. 

The formula we use for a complex number is a + bi, where:

• 'a' and 'b' are real numbers
• 'i' is an imaginary number equal to the square root of – 1
• 'a' is the real part
• 'b' is the imaginary part

The complex conjugate of a + bi is a – bi, where 'a' is the real part and the imaginary part is –b.
 

The conjugate of a complex number can be mathematically denoted as z. Since z and z are each other’s conjugates, they are together called a complex-conjugate pair.
Consider \(z = x + iy \) as a complex number, so the conjugate of z can be represented as:
\(z = x – iy\)
The relationship between z and z shows us that the conjugate is found by simply changing the sign of the imaginary part.

In geometry, the conjugate of a complex number z is its reflected or mirrored version across the x-axis in the complex plane, also called the Argand plane, as shown below:

Multiplication of complex number with its conjugate

Multiplying a complex number by its complex conjugate always results in a real number. 
For example, We multiply the complex number a + bi by its conjugate a – bi
\( (a + bi)(a – bi) = a² – (b²)(i²) = a² + b²\)

The complex conjugate of a matrix A with complex entries is a matrix in which each element is replaced by its complex conjugate. For example, consider the row matrix \(A=[2+3i, 5−4i, 1+7i].\) Its complex conjugate is \(B=[2−3i, 5+4i, 1−7i]\), where each entry in B is the conjugate of the corresponding entry in A. The complex conjugate of A is denoted by \(\overline{A}\), so \(B = \overline{A} \). This demonstrates how we can find the complex conjugate for any matrix with complex entries.

If z, z1, and z2 are complex numbers, their properties are as follows:

When the conjugate of a complex number is equal to the number itself, it indicates the number is real. We represent it as: z = z. 
For example: \((5 + 0i) = (5 – 0i) = 5.\)

If the sum of a complex number and its conjugate is zero, the number is considered to be imaginary. This can be mathematically expressed as: \((z + z = 0)\).

For example: the conjugate of \((0 – 5i) = (0 + 5i) = 5i\)

\((z ) = z\)

\(z +  z = 2 Re (z)\)

\(z – z = 2i Im(z)\)

\( z \cdot \overline{z} = (\text{Re}(z))^2 + (\text{Im}(z))^2 \)

\( (z_1 + z_2) = z_1 + z_2 \)

\( (z_1 - z_2) = z_1 - z_2 \)

\( (z_1 \times z_2) = z_1 z_2 \)

\( z = \frac{z_1}{z_2}, \quad z_2 \neq 0 \)

Keep in mind that the conjugate of a complex number can only be found if the complex number is in the form:

\(z = (x + iy)\). If not, convert it to this form before finding the conjugate.
 

The complex conjugate root theorem states that if p(x) is a polynomial with real coefficients and \(a + bi\) as its root, then \(a – bi\) is also a root of the polynomial.

Complex conjugates play a vital role in different fields. Students must understand how complex conjugates simplify expressions. Here are some applications:

• Complex conjugates are widely utilized in studying alternating current circuits. For example, engineers apply complex conjugate in calculating impedance, current, and voltage.

• They are used in aerodynamics for structuring airplane wings and analyzing the airflow around objects.

• In the medical field, they enhance image quality and assist in detecting medical conditions in MRI scans.

• We use complex conjugates to analyze sound waves and other mechanical vibrations. 

• They are utilized in signal processing in mobile networks to enhance the transmission of messages.