Sylvester's theorem provides a rapid way to calculate functions of a matrix. Some simple functions of a matrix of frequent occurrence are and (for N large). Two more matrix functions which are very important in wave propagation are and .Before going into the somewhat abstract proof of Sylvester's theorem, we will take up a numerical example. Consider the matrix
Let us consider the behavior of the matrix .
Any power of this matrix is the matrix itself, for example its square.
This property is called idempotence (Latin for self-power). It arises because .The same thing is of course true of . Now notice that the matrix is ``perpendicular'' to the matrix , that is
since and are perpendicular.
Sylvester's theorem says that any function f of the matrix A may be written
The simplest example is
The inverse is
The identity matrix may be expanded in terms of the eigenvectors of the matrix A.
Before illustrating some more complicated functions let us see what it takes to prove Sylvester's theorem. We will need one basic result which is in all the books on matrix theory, namely, that most matrices (see exercises) can be diagonalized. In terms of our example this takes the form
Using the orthonormality of and this reduces to
It is clear how (36) can be used to prove Sylvester's theorem for any polynomial function of A. Clearly, there is nothing peculiar about matrices either. This works for .Likewise, one may consider infinite series functions in A. Since almost any function can be made up of infinite series, we can consider also transcendental functions like sine, cosine, exponential.
Exponentials arise naturally as the solutions to differential equations. Consider the matrix differential equation
This is the power series definition of an exponential function. If the matrix A is one of that vast majority which can be diagonalized, then the exponential can be more simply expressed by Sylvester's theorem. For the numerical example we have been considering, we have
The exponential matrix is a solution to the differential equation (37) without regard to boundaries. It frequently happens that physics gives one a differential equation
is the solution to (38) for arbitrary constants k1 and k2. Boundary conditions are then used to determine the numerical values of k1 and k2. Note that k1 and k2 are just y2 (x = 0) and y2 (x = 0).
An interesting situation arises with the square root of a matrix. A matrix like A will have four square roots because there are four possible combinations for choice of plus or minus signs on and .In general, an matrix has 2n square roots. An important application arises in a later chapter, where we will deal with the differential operator .The square root of an operator is explained in very few books and few people even know what it means. The best way to visualize the square root of this differential operator is to relate it to the square root of the matrix M where
The right-hand matrix is a second difference approximation to a second partial derivative. Let us define
Clearly we wish to consider M generalized to a very large size so that the end effects may be minimized. In concept, we can make M as large as we like and for any size we can get square roots. In practice there will be only two square roots of interest, one with the plus roots of all the eigenvalues and the other with all the minus roots. How can we find these ``principal value'' square roots? An important case of interest is where we can use the binomial theorem so that
provided that k is not a function of x. If k is a function of x, the square root of the differential operator still has meaning but is not so simply computed with the binomial theorem.
as to see why one eigenvector is lost. This is called a defective matrix. (This example is from T. R. Madden.)
What is the x dependence of the solution when ab is positive? When ab is negative? Assume a and b are independent of x. Use Sylvester's theorem. What would it take to get a defective matrix? What are the solutions in the case of a defective matrix?