Page 412 - 35Linear Algebra

412 Movie Scripts

where dim V = n < ∞. Since V is finite dimensional, we can represent L by a
square matrix M by choosing a basis for V .
So the eigenvalue equation

Lv = λv

becomes

Mv = λv,

where v is a column vector and M is an n×n matrix (both expressed in whatever
basis we chose for V ). The scalar λ is called an eigenvalue of M and the job
of this video is to show you how to find all the eigenvalues of M.
The first step is to put all terms on the left hand side of the equation,
this gives
(M − λI)v = 0 .
Notice how we used the identity matrix I in order to get a matrix times v
equaling zero. Now here comes a VERY important fact

Nu = 0 and u 6= 0 ⇐⇒ det N = 0.

I.e., a square matrix can have an eigenvector with vanishing eigenvalue if and only if its
determinant vanishes! Hence

det(M − λI) = 0.

The quantity on the left (up to a possible minus sign) equals the so-called
characteristic polynomial

P M (λ) := det(λI − M) .
It is a polynomial of degree n in the variable λ. To see why, try a simple
2 × 2 example

a b λ 0 a − λ b
det − = det = (a − λ)(d − λ) − bc ,
c d 0 λ c d − λ
which is clearly a polynomial of order 2 in λ. For the n × n case, the order n
term comes from the product of diagonal matrix elements also.
There is an amazing fact about polynomials called the fundamental theorem
of algebra: they can always be factored over complex numbers. This means that

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