Understanding Orthogonal & Linear Combinations

[touqra]

I was reading the literature, and come across this. But I couldn't decipher what it means by linear combination and orthogonal combination.

It is sufficient to consider two axionic fields, A and B, with a potential:

$$V = \lambda_{1}^4 [1 - cos(\frac{\theta}{f_1} + \frac{\rho}{g_1})] + \lambda_{2}^4 [1 - cos(\frac{\theta}{f_2} + \frac{\rho}{g_2})]$$

It is easy to see that, when the condition

$$\frac{f_1}{g_1} = \frac{f_2}{g_2}$$

is met, the same linear combination of the two axions appears in both terms. Hence, the orthogonal combination is a flat direction of V.

 

[LightHero]

Linear combination and orthogonal combination refer to the mathematical concept of linear algebra. A linear combination is a sum of vectors, where each vector is multiplied by a scalar value. In this example, the two axions A and B are being combined in a linear way to create a new vector (the linear combination). The orthogonal combination is then a vector that is perpendicular (orthogonal) to the linear combination.

 

[sir-pinski]

In simpler terms, a linear combination is a mathematical operation where two or more values are multiplied by a constant and then added together. In the given example, the potential function is a linear combination of two axionic fields, A and B, where the values are multiplied by constants (lambda) and then added together.

On the other hand, an orthogonal combination is a combination of values that are perpendicular or at right angles to each other. In the given example, the condition for an orthogonal combination to exist is when the ratio of the constants (f/g) for both axionic fields is equal. This means that the two axionic fields are at right angles to each other and their combination results in a flat direction in the potential function.

Overall, understanding orthogonal and linear combinations is important in mathematics and physics as it allows us to manipulate and analyze complex functions and equations. In this example, it helps us identify a flat direction in the potential function, which can have implications in the behavior of the system being studied.