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Complex numbers, conjugate complex numbers,

complex conjugate matrices

Complex number. Any number of the form a + bi , where a and b are real numbers and

i2 = -1.

Equality of two complex numbers. Two complex numbers are defined to be equal if and only if they are identical.

Conjugate of a complex number. The conjugate of the complex number z = a + bi is a - bi . Two numbers of the type a + bi and a - bi, where a and b are real, are called conjugate complex numbers. We denote the conjugate of a complex number z by . The conjugate of a complex number is also called the complex conjugate.

Example. The numbers 5 + 2i and 5 - 2i are conjugates.

A real number is its own conjugate.

Absolute value of a complex number. The absolute value |z| of the complex number z = a + bi is given by

.

A complex number a + bi = 0 if and only if a = b = 0.

Laws for conjugate complex numbers.

1] The conjugate of the conjugate of a complex number is the number itself:

2] The conjugate of the sum of two complex numbers is the sum of their conjugates:

3] The conjugate of the product of two complex numbers is the product of their conjugates:

Conjugate of a matrix. The matrix whose elements are the complex conjugates of the corresponding elements of the given matrix. The conjugate of a matrix is also called the complex conjugate.

Example. Let A be given by

Then the conjugate of A, denoted by and called “A conjugate”, is

Laws for conjugate matrices. Let A and B be matrices and k be a scalar. The following hold for matrices and their conjugates:

1] The conjugate of the conjugate of a matrix is the matrix itself:

2] The conjugate of k∙A is the conjugate of k times the conjugate of A:

3] The conjugate of the sum of two matrices is the sum of their conjugates:

4] The conjugate of the product of two matrices is the product of their conjugates:

5] The transpose of the conjugate of a matrix is the conjugate of the transpose: