Assign the value of the variable a to represent an mn matrix with name "a":

Arithmetic operations recognize that a is not a scalar:

D recognizes that a is a matrix variable:

Compute derivatives with respect to a matrix variable:

Compute derivatives involving matrix-valued functions:

This requires a real-valued matrix:

Use a matrix variable in optimization:

Solve equations and inequalities involving a matrix variable:

Approximate the determinant of a perturbed matrix:

Order zero approximation:

Compare with the exact value:

Order one approximation:

Compare with the exact value:

Order two approximation:

Compare with the exact value:

Derive a least-squares solution for data given as a list of pairs :

Find the vector of vertical deviations for the data:

Define the sum of squares of the vertical deviations for the data:

Set up the least-squares equations:

Generate some data:

Solve the least-squares problem for this data:

Find an optimality condition for a portfolio optimization problem with the expected return and standard deviation :

The goal is to maximize when the vector of asset weights satisfies Total[x]=1. The constraint can be used to represent where the unconstrained vector variable consists of the first coordinates of :

The maximum occurs at a critical point of :

Express the condition in terms of :

Compute the gradient of the log-likelihood function of the linear regression model represented by the equation , where are normally distributed random variables with mean zero and variance :

The log-likelihood function is given by:

Compute :

Express the result in terms of :

Compute :