probably to identify whichever layout is used and maintain consistency with it, rather than attempting to use the same layout in all situations. Alternatives 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 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. NOTE: 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. Vector-by-scalar The derivative of a vector , by a scalar x is written (in numerator layout notation) as In vector calculus the derivative of a vector y with respect to a scalar x is known as the tangent vector of the vector y, . Notice here that y: R1 → Rm. Example Simple examples of this include the velocity vector in Euclidean space, which is the tangent vector of the position vector (considered as a function of time). Also, the acceleration is the tangent vector of the velocity. Scalar-by-vector The derivative of a scalar y by a vector , is written (in numerator layout notation) as In vector calculus, the gradient of a