Partial Differential Equations: Analytical and Numerical Methods
Emphasizing techniques over theory, this book focuses its discourse mainly on physical and mathematical derivations of Laplace, wave, and heat equations.
TABLE OF CONTENTS Partial Differential Equations: Analytical and Numerical Methods
Appendix A: Proof of Theorem 3.47
Theorem A.1
Let A ? R n n be symmetric, and suppose A has an eigenvalue ? of (algebraic) multiplicity k (meaning that ? is a root of multiplicity k of the characteristic polynomial of A). Then A has k linearly independent eigenvectors corresponding to ?.
| Proof. | We argue by induction on n. The result holds trivially for n = 1. We assume it holds for symmetric ( n ? 1) ( n ? 1) matrices, and suppose A ? R n n is symmetric and has an eigenvalue ? of (algebraic) multiplicity k. There exists x 1 ? 0 such that Ax 1 = ? x 1. We can assume that x 1 = 1 (since x 1/ x 1 is also an eigenvector of A), and we extend x 1 to an orthonormal basis { x 1, x 2, , x n}. Then  where X ? R n n is defined by X = [ x 1 x 2 x n]. We have  and  (The symbol ? ij is 1 if i = j and 0 if i ? j.) Therefore,  where B ? R (n ?1) (n ?1) is defined by  The matrix B is symmetric. Also,  since X T = X ? |
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