Moduluskvadrat

Moduluskvadrat, also known as the modulus squared or the squared modulus, is a mathematical concept that applies to complex numbers and vectors. It is defined as the product of a number with its complex conjugate. For a complex number z = a + bi, where a and b are real numbers, and i is the imaginary unit, the modulus squared is given by |z|^2 = z * z* = (a + bi)(a - bi) = a^2 + b^2. This concept is widely used in various fields of mathematics and physics, including signal processing, quantum mechanics, and electrical engineering.

In the context of vectors, the modulus squared is the dot product of a vector with itself. For a vector v = (v1, v2, ..., vn), the modulus squared is given by |v|^2 = v · v = v1^2 + v2^2 + ... + vn^2. This concept is fundamental in linear algebra and is used to calculate the magnitude or length of a vector.

The modulus squared is always a non-negative real number, as it is the sum of squares of real numbers. It is also invariant under rotations and reflections, meaning that it remains unchanged under these transformations. This property makes it a useful tool in various applications, such as calculating the power of a signal in signal processing or the probability amplitude in quantum mechanics.