Engineering Analysis/Matrices

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Norms[edit]

Induced Norms[edit]

n-Norm[edit]

Frobenius Norm[edit]

Spectral Norm[edit]

Derivatives[edit]

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-Inverses[edit]

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-Inverse[edit]

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-Inverse[edit]

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