Here is a short read on this issue.Logarythmic said:A question for my understanding:
If I have an operator [tex]\cal{L}[/tex] and a set of eigenfunctions [tex]\phi_n[/tex] of this operator, then the eigenfunctions are orthogonal. Why is that?
Orthogonal functions are a set of mathematical functions that are perpendicular to each other when graphed. This means that when plotted on a graph, the functions intersect at a right angle. They are commonly used in fields such as physics, engineering, and signal processing.
Orthogonal functions are used in science to model and analyze complex systems. They allow scientists to break down a complex function into simpler, orthogonal components, making it easier to understand and manipulate. They are also used in statistical analysis to determine relationships between variables.
Understanding orthogonal functions is important because they are a fundamental concept in mathematics and science. They are used in a wide range of applications and can help simplify complex problems. Additionally, many mathematical techniques and algorithms rely on the concept of orthogonality, so understanding it is crucial for further scientific advancements.
Yes, orthogonal functions can be non-linear. While linear functions are commonly used in orthogonal transformations, non-linear functions can also be orthogonal if they satisfy the conditions of orthogonality. This means that they must be perpendicular to each other when graphed.
Orthogonal functions are different from orthonormal functions in that orthonormal functions not only satisfy the conditions of orthogonality, but they also have a magnitude of 1. This means that orthonormal functions are not only perpendicular to each other, but they also have the same "length". In other words, the area under the curve of an orthonormal function is equal to 1, while for orthogonal functions it can be any value.