San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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The Extention of Calculus Operations to Matrices |
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The calculus operation of taking derivatives when applied to matrix functions is a whole order of complexity greater than that of ordinary calculus. Take for example the simplest of matrix functions
The increments ΔN are given in terms of the increments ΔM by
As the increments ΔM reduce to a matrix of infinitesimals dM the relation reduces to
This is quite different from what might be expected in analogy with the calculus of real variables; i.e.,
Something of this sort can be achieved with the use of the commutator function for matrices,
Thus dM·M is equal to MdM − [M, dM] and
What immediately is clear is that derivatives involving matrices are awkward. Something in the nature of (∂N/∂M) as a matrix would be an n²×n² matrix consisting of n² blocks each of which represent (∂n_{ij}/∂M). It would be better to consider (∂N/∂M) as a four dimensional object called a tensor. But better yet the relationship between the changes in N due to changes in M should be displayed, as above, in terms of the relationship between the differentials dN and dM.
We see that if P=M³ then
The generalization to Q=M^{k} is obvious though awkward.
The useful matrix exponential function is defined as
where I is the n×n identity matrix.
Clearly the increment in the exponential function exp(M+ΔM)−exp(M) is going to be complicated even when the increments in M are infinitesimal.
For the special case of a ΔM that commutes with M
(To be continued.)