Sesquilinear form

Alright kiddo, let me explain to you what a sesquilinear form is!

First of all, do you know what a linear function is? It's a function where if you double the input, you double the output. For example, if you have a function f(x) = 2x, if you put in 3, you get out 6, but if you put in 6, you get out 12.

Now, a sesquilinear form is kind of like a linear function, but not exactly. It's a function that takes in two inputs, and it's kind of like a multiplication or dot product of those two inputs. Let's call the two inputs x and y.

So, a sesquilinear form looks like this: f(x,y) = x * A * y, where A is just some number or matrix.

The difference between a linear function and a sesquilinear form is that with a sesquilinear form, when you scale one of the inputs (let's say x), the output doesn't necessarily just scale by the same amount. Instead, it scales by the complex conjugate of that amount, which is a fancy way of saying it flips the sign of any imaginary part.

So, if you multiply x by 2, the output doesn't just double, it doubles and potentially flips the sign of its imaginary part.

This might sound a bit complicated, but the important thing to remember is that a sesquilinear form is just a special kind of function that takes in two inputs and kind of multiplies them together, but not exactly like a linear function.