Engineering Analysis/Matrices

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NormsEdit

Induced NormsEdit

n-NormEdit

Frobenius NormEdit

Spectral NormEdit

DerivativesEdit

Consider 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-InversesEdit

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:

 

Right Pseudo-InverseEdit

Consider 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-InverseEdit

Consider the following matrix:

 

We call L the left pseudo-inverse of A because

 

but

 

We will denote the left pseudo-inverse of A as