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Notation[ edit ] The vector and matrix derivatives presented in the sections to follow take full advantage of matrix notationusing a single variable to represent a large number of variables. In what follows we will distinguish scalars, vectors and matrices by their typeface.
Such matrices will be denoted using bold capital letters: A, X, Y, etc. An element of M n,1that is, a column vectoris denoted with a boldface lowercase letter: An element of M 1,1 is a scalar, denoted with lowercase italic typeface: XT denotes matrix transposetr X is the traceand det X or X is the determinant.
All functions are assumed to be of differentiability class C1 unless otherwise noted. Generally letters from the first half of the alphabet a, b, c, … will be used to denote constants, and from the second half t, x, y, … to denote variables.
As mentioned above, there are competing notations for laying out systems of partial derivatives in vectors and matrices, and no standard appears to be emerging yet. The next two introductory sections use the numerator layout convention simply for the purposes of convenience, to avoid overly complicating the discussion.
The section after them discusses layout conventions in more detail. It is important to realize the following: Despite the use of the terms "numerator layout" and "denominator layout", there are actually more than two possible notational choices involved.
The reason is that the choice of numerator vs. The choice of numerator layout in the introductory sections below does not imply that this is the "correct" or "superior" choice. There are advantages and disadvantages to the various layout types.
Serious mistakes can result from carelessly combining formulas written in different layouts, and converting from one layout to another requires care to avoid errors. As a result, when working with existing formulas the best policy is probably to identify whichever layout is used and maintain consistency with it, rather than attempting to use the same layout in all situations.
Alternatives[ edit ] The tensor index notation with its Einstein summation convention is very similar to the matrix calculus, except one writes only a single component at a time. It has the advantage that one can easily manipulate arbitrarily high rank tensors, whereas tensors of rank higher than two are quite unwieldy with matrix notation.
All of the work here can be done in this notation without use of the single-variable matrix notation. However, many problems in estimation theory and other areas of applied mathematics would result in too many indices to properly keep track of, pointing in favor of matrix calculus in those areas.
Also, Einstein notation can be very useful in proving the identities presented here see section on differentiation as an alternative to typical element notation, which can become cumbersome when the explicit sums are carried around.
Note that a matrix can be considered a tensor of rank two. Derivatives with vectors[ edit ] Main article: Vector calculus Because vectors are matrices with only one column, the simplest matrix derivatives are vector derivatives. The notations developed here can accommodate the usual operations of vector calculus by identifying the space M n,1 of n-vectors with the Euclidean space Rn, and the scalar M 1,1 is identified with R.
The corresponding concept from vector calculus is indicated at the end of each subsection. The discussion in this section assumes the numerator layout convention for pedagogical purposes.
Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.
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