The Big Eigenvalue - Eigenvector Theorem

Definition:  An nxn matrix A is said to be diagonalizable if there exists an nxn diagonal matrix D and an nxn invertible matrix P such that A = PDP^-1 .

Theorem: An nxn matrix A is diagonalizable if and only if Rⁿ has a basis of eigenvectors of A, and, in fact, when this happens, if we set P to be the matrix whose columns are these eigenvectors and D the diagonal matrix whose entries are the corresponding eigenvalues, then A = PDP^-1 .

Theorem: [The Big Eigenvalue - Eignevector Theorem]

Let A be an nxn matrix.

 

Let p(λ) = det (A - λI).  That is, p(λ) is the characteristic polynomial of A.

Then λ

 

(1) The degree of p(λ) is exactly n.

 

(2) p(λ) has at most n roots.

 

Assume, further, that   p(λ) factors in the following form:

where the λ_i's are distinct.  [Recall, m_i is called the multiplicity of λ_i.]

Then

(3) m₁ + m₂ + ... + m_p = n

 

(4) p = n if and only if each m_i = 1, and in this case A is diagonalizable.  In fact, if v_i is an eigenvector of λ_i, and if we set P to be the nxn matrix whose columns are these v_i's and if we set D to be the nxn diagonal matrix whose diagonal entries are the λ_i's, then, A = PDP^-1.  [Recall, that if a collection of eigenvectors have distinct eigenvalues, then they are linearly independent.]

 

(5) dim {v | Av = λ_iv } < m_i, for each i.   [Recall:  {v | Av = λ_iv } is called the eigenspace of v_i.]

 

(6)         A is diagonalizable

 

 if and only if

 

 dim (eigenspace of v_i ) = m_i for EACH i.

if and only if

 

the sum of the dimension of the eigenspaces of A = n

 

AND, in this case a basis of Rⁿ which are eigenvectors for A can be constructed by finding a basis for each eigenspace of A and combining (taking the union of) these eigenvectors to form one large set.  Furthermore, the P and D in the formula A = PDP^-1 are found as follows:  P is the matrix whose columns are that basis of eigenvectors of Rⁿ and D is the diagonal matrix which has m₁ copies of λ₁, , m₂ copies of λ₂, ...., m_p copies of λ_p.

 

A Useful formula:  If a matrix is triangular (that is, is either filled with zeros below the main diagonal or filled with zeros above the main diagonal) then its determinant is the product of the diagonal entries of this matrix.