# Orthogonal Functions: Questioning the Reason Why

• Logarythmic
In summary, orthogonal functions are a set of mathematical functions that intersect at a right angle when graphed. They are used in various fields of science to model and analyze complex systems, and are also important in statistical analysis. Non-linear functions can also be orthogonal if they satisfy the conditions of orthogonality. Orthonormal functions are similar to orthogonal functions, but also have a magnitude of 1, making them "normalized".
Logarythmic
A question for my understanding:

If I have an operator $$\cal{L}$$ and a set of eigenfunctions $$\phi_n$$ of this operator, then the eigenfunctions are orthogonal. Why is that?

Logarythmic said:
A question for my understanding:

If I have an operator $$\cal{L}$$ and a set of eigenfunctions $$\phi_n$$ of this operator, then the eigenfunctions are orthogonal. Why is that?
Here is a short read on this issue.

http://vergil.chemistry.gatech.edu/notes/quantrev/node16.html

## 1. What are orthogonal functions?

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.

## 2. How are orthogonal functions used in science?

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.

## 3. Why is it important to understand orthogonal functions?

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.

## 4. Can orthogonal functions be non-linear?

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.

## 5. How are orthogonal functions different from orthonormal functions?

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.

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