# Reference for matrix calculus

Could someone provide a good reference for learning matrix calculus? I've recently moved to a more engineering-oriented field where it's commonly used and don't have much experience with it.

• Could you be a bit more specific? What exactly are you trying to learn and what do you already know? Do you want to understand things such as A = exp(Ct) solves dA/dt = CA for A, C matrices? Or perhaps even study matrix Lie groups and their differential properties (I can imagine this could have many engineering applications)? Nov 11, 2010 at 19:46
• Well, in linear regression, one can derive a closed-form solution in the form of the normal equations: en.wikipedia.org/wiki/Normal_equations#General_linear_model . The derivation uses results about the gradient of the trace of a matrix product. I guess I just wanted to feel comfortable with these ideas - maybe build up a geometric picture and get some insight as to how everything works. Nov 11, 2010 at 19:55

Actually the books cited above by Sivaram are excellent for numerical stuff. If you want "matrix calculus" then the following books might be helpful:

1. Matrix Differential Calculus with Applications in Statistics and Econometrics by Magnus and Neudecker

2. Functions of Matrices by N. Higham

3. Calculus on Manifolds by Spivak

Some classic, but very useful material can also be found in

1. Introduction to Matrix Analysis by Bellman.

As a simple example, the books will teach (unless you already know it) how to compute, say, the derivative of $f(X) = \log\det(X)$ for an invertible matrix $X$.

Gene Howard Golub, Charles F. Van Loan book on "Matrix Computations" is regarded as the "Bhagavad Gita" for Matrix Algorithms.