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

(26) |

(27) | ||

(28) |

(29) | ||

(30) |

(31) | ||

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

(32) |

(33) |

(34) |

(35) |

(36) |

Exponentials arise naturally as the solutions to differential equations. Consider the matrix differential equation

(37) |

(38) |

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 2^{n} 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

- Premultiply (31)b by and postmultiply (31)c by ,then subtract. Is a necessary condition for and to be perpendicular? Is it a sufficient condition?
- Show the Cayley-Hamilton theorem, that is, if then
- Verify that, for a general matrix
**A**, for which**A**. What is the general form for ? - For a symmetric matrix it can be shown that there is always a complete set of eigenvectors. A problem sometimes arises with nonsymmetric matrices. Study the matrix as to see why one eigenvector is lost. This is called a defective matrix. (This example is from T. R. Madden.)
- A wide variety of wave-propagation problems in a stratified medium
reduce to the equation
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? - Consider a matrix of the form where
**v**is a column vector and is its transpose. Find in terms of a power series in .[Note that collapses to times a scaling factor, so the power series reduces considerably.] - The following ``cross-product'' matrix often arises in electrodynamics.
Let
**(a)**- Write out elements of .
**(b)**- Show that or .
**(c)**- Let be an arbitrary vector. In what geometrical
directions do
**Uv**, , and point? **(d)**- What are the eigenvalues of
**U**? [Hint: Use part (b).] **(e)**- Why cannot
**U**be canceled from ? **(f)**- Verify that the idempotent matrices of
**U**are

10/30/1997