Linear Algebra Question (Goldstein Classical Mechanics Chapter 8 Hamilton's Equations Of Motion)

Posted on 2008-03-14 23:08 By dcasimir ( Maple Rating 3

296)

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Math Advice (Student)

Show that if lambda[i] are the eigenvalues of a square matrix, then if the reciprocal matrix exists it has the eigenvalues (lambda[i])^(-1).

By "reciprocal matrix" I assume he means inverse.

outline

Comment on 2008-03-15 00:51 from acer ( Maple Rating 4

1410)

Start off with the definition of an eigenvalue. If scalar lambda[i] is an eigenvalue then there exists a nontrival (not all-zero) vector x such that,

    A . x = lambda[i] * x

Now, supposing that A^-1 (the inverse of A) exists, multiply (from the left) both sides of that equation by A^-1. This should result in,

    x = A^-1 . lambda[i] . x

Now, multiply both sides of that equation on the left by scalar 1/lambda[i]. Realize that on the right-hand-side of the equation the scalar lambda[i] can be pulled out to the front (using linearity properties) allowing it to cancel with the 1/lambda[i] multiplicative factor. This should result in,

   1/lambda[i] * x = A^-1 . x

Interpret that according to the definition of an eigenvalue. Add a note about whether lambda[i] can be zero.

acer

Thanks

Comment on 2008-03-15 12:21 from dcasimir ( Maple Rating 3

296)

Thanks a lot.

I'm embarrassed. I bet this is a standard example in some Linear Algebra text somewhere.

dc.