# Engineering Analysis/Matrices

## Norms

edit### Induced Norms

edit### n-Norm

edit### Frobenius Norm

edit### Spectral Norm

edit## Derivatives

editConsider the following set of linear equations:

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

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

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

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

editThere 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:

### Right Pseudo-Inverse

editConsider the following matrix:

We call this matrix R the **right pseudo-inverse** of A, because:

but

We will denote the right pseudo-inverse of A as

### Left Pseudo-Inverse

editConsider the following matrix:

We call L the **left pseudo-inverse** of A because

but

We will denote the left pseudo-inverse of A as