Engineering Analysis/Matrices

Norms

Induced Norms

n-Norm

Frobenius Norm

Spectral Norm

Derivatives

Consider the following set of linear equations:

$a = bx_1 + cx_2$

$d = ex_1 + fx_2$

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

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

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

$x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$

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

$B = Ax$

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

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

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:

$AB = BA = I$

Right Pseudo-Inverse

Consider the following matrix:

$R = A^T[AA^T]^{-1}$

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

$AR = I$

but

$RA \ne I$

We will denote the right pseudo-inverse of A as $A^\dagger$

Left Pseudo-Inverse

Consider the following matrix:

$L = [A^TA]^{-1}A^T$

We call L the left pseudo-inverse of A because

$LA = I$

but

$AL \ne I$

We will denote the left pseudo-inverse of A as $A^\ddagger$