Matrix calculus

If the eigenvalues of [X] are distinct so that the characteristic equation of [X] does not 

For a matrix of order 2, a possible modal matrix formed from the eigenvectors (Eq. A.4.17 and A.17.18) is 

Other structures for [ ] are possible it all depends on what values we choose for and 

Polynomials and exponentials play an important role in matrix calculus and matrix dilferen-tial equations. It is, therefore, necessary to develop techniques for calculating these functions. 

If we recall from calculus that many functions can be written as a Maclaurin series, then 

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Walker, M. J., 1954. Matrix calculus and the Stokes parameters of polarized radiation, Am. J. Phys., 22, 170-174. 

As to the practical way of producing a suitable modulation, a series of proposals have been made. In practically all cases a modulator is introducted into the light beam, whereas the polarizing prisms are kept at rest. It appears that the general shape of the lower curve in Fig. 6.5 remains unchanged. More precise calculations can be carried out with the aid of a matrix calculus, as reviewed by Walker (216). 

E. Bodewig, Matrix Calculus , North-Holland, Amsterdam, 1959, p. 67. 

The appearance of Stokes publication, in which the concept of a vector representation of the beam description parameters was introduced, renders back to 1852. It, however, has remained for many years unnoticed (Shurcliff, 1962). The need for a general systematic and effective procedure for solving the exponentially growing quantity of optical problems has led to the rediscovery of Stokes approach to polarized radiation. Mueller s contribution consists of developing a matrix calculus for evaluation of the four elements of the Stokes vector which are a set of quantities describing the intensity and polarization of a light beam. 

Some of the reasoning in the book uses matrix formalism. This is for the sake of convenience, since some quantitative relations are more easily expressed in matrix language than otherwise. Readers without any previous experience of matrix calculus may skim those paragraphs in which matrix calculus is used without losing too much... 

The model can also be written in matrix notation as below. It is very convenient to use matrices to describe the calculations involved in the modelling process. A short summary of matrix calculus is given in an appendix at the end of this book. 

In matrix notation (see Appendix Matrix calculus), the above expression can be written... 

The same example as above is used to illustrate the computations. A brief summary of matrix calculus is given in Appendix Matrix calculus at the end of this book. The whole series of experiments can be summarized by the now well-known matrix relation... 

A very readable, and amusing , book on matrix calculus is Gilbert Strang... 

This is the same result as obtained by the use of eq. (2.5), however, applying matrix calculus with less expenditure. The only disadvantage is the matrix inversion, but modem software provides an easy way to do this (see software MAPLE used in Appendix 6.1). 

Depending on the circumstances, it makes sense to apply either tensor or matrix calculus. Occasionally it may be useful to switch the representation. Typically, the results of a derivation requiring tensors are written in the more accessible matrix form. While operations involving scalars and vectors are applicable for both, the more general case is subjected to restrictions ... 

Usually within the framework of matrix calculus, the vector operations are retained or may be replaced with pure matrix algebra. In matrix notation, the scalar product of two vectors may be represented by the matrix product of a row and a column matrix ... 

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