Norms

Derivatives

Consider the following set of linear equations:

${\displaystyle a=bx_{1}+cx_{2}}$

${\displaystyle d=ex_{1}+fx_{2}}$

We can define the matrix A to represent the coefficients, the vector B as the results, and the vector x as the variables:

${\displaystyle A={\begin{bmatrix}b&c\\e&f\end{bmatrix}}}$

${\displaystyle B={\begin{bmatrix}a\\d\end{bmatrix}}}$

${\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}$

And rewriting the equation in terms of the matrices, we get:

${\displaystyle B=Ax}$

Now, let's say we want the derivative of this equation with respect to the vector x:

${\displaystyle {\frac {d}{dx}}B={\frac {d}{dx}}Ax}$

We know that the first term is constant, so the derivative of the left-hand side of the equation is zero. Analyzing the right side shows us:

Pseudo-Inverses

There are special matrices known as pseudo-inverses, that satisfies some of the properties of an inverse, but not others. To recap, If we have two square matrices A and B, that are both n × n, then if the following equation is true, we say that A is the inverse of B, and B is the inverse of A:

${\displaystyle AB=BA=I}$