The characteristic polynomial

How can eigenvalues be found without guessing? By turning the equation Av = λv into an equation in λ alone.

From the eigen-equation to a determinant

A v = λ v
A v - λ v = 0
(A - λI) v = 0            <- careful: λI, not λ on its own

This last line says v lies in the kernel of A - λI. But we want a non-zero v: that kernel must therefore contain more than zero, meaning the matrix A - λI must fail to be invertible. And that is tested by the determinant:

λ is an eigenvalue   <=>   det(A - λI) = 0

The characteristic polynomial

The quantity det(A - λI) is a polynomial in λ, of degree n for an n × n matrix. It is called the characteristic polynomial, and its roots are exactly the eigenvalues.

On an example:

A = [ 3  1 ]      A - λI = [ 3-λ    1  ]
    [ 2  2 ]               [  2    2-λ ]

det = (3-λ)(2-λ) - 1×2 = 6 - 3λ - 2λ + λ² - 2 = λ² - 5λ + 4

λ² - 5λ + 4 = 0   ->   λ = 1   or   λ = 4

Finding the eigenvectors

Once λ is known, solve the homogeneous system for each one:

For λ = 4 :  (A - 4I)X = 0
             [ -1   1 ] [x]   [0]        -x + y = 0   ->   y = x
             [  2  -2 ] [y] = [0]

             E(4) = Vect( (1 ; 1) )

For λ = 1 :  (A - I)X = 0
             [ 2  1 ] [x]   [0]          2x + y = 0   ->   y = -2x
             [ 2  1 ] [y] = [0]

             E(1) = Vect( (1 ; -2) )

The check, always quick and always worth doing:

A(1 ; 1)  = (3+1 ; 2+2)   = (4 ; 4)   = 4 × (1 ; 1)    ✔
A(1 ; -2) = (3-2 ; 2-4)   = (1 ; -2)  = 1 × (1 ; -2)   ✔

Two free sanity checks

Two relations let you verify eigenvalues without recomputing anything:

sum of the eigenvalues     = trace(A)   (sum of the diagonal)
product of the eigenvalues = det(A)

On the example: 1 + 4 = 5 = 3 + 2 ✔ and 1 × 4 = 4 = 6 - 2 ✔. In dimension 2 these two relations are even enough to find the eigenvalues in your head.

Multiplicities and special cases

A root may be repeated. For λ of multiplicity m in the characteristic polynomial:

1 ≤ dim E(λ) ≤ m

When dim E(λ) < m, eigenvectors are "missing" — precisely what will prevent diagonalisation. Two cases can be read at a glance:

- TRIANGULAR or diagonal matrix: the eigenvalues sit on the DIAGONAL.
- real symmetric matrix: all eigenvalues are real
  (and the matrix is always diagonalisable — the spectral theorem).

Summary

• λ is an eigenvalue ⟺ det(A - λI) = 0.
• That determinant is the characteristic polynomial, of degree n.
• Its roots are the eigenvalues; the eigenvectors then come from (A - λI)X = 0.
• Checks: sum = trace, product = determinant.
• Triangular matrix: the eigenvalues are on the diagonal.
• Real symmetric matrix: real eigenvalues, diagonalisation guaranteed.