Eigenvalues & Eigenvectors (DP IB Maths: AI HL)

Eigenvalues and eigenvectors are properties of square matrices and are used in a lot of real-life applications including geometrical transformations and probability scenarios. In order to find these eigenvalues and eigenvectors, the characteristic polynomial for a matrix must be found and solved.

What is a characteristic polynomial?

• For a matrix , if when  is a non-zero vector and  a constant, then  is an eigenvalue of the matrix  and  is its corresponding eigenvector
• If or and for to be a non-zero vector, 
• The characteristic polynomial of an  matrix is:

• In this course you will only be expected to find the characteristic equation for a matrix and this will always be a quadratic

How do I find the characteristic polynomial?

• STEP 1
  Write , remembering that the identity matrix must be of the same order as 
• STEP 2
  Find the determinant of  using the formula given to you in the formula booklet

• STEP 3
  Re-write as a polynomial 

How do you find the eigenvalues of a matrix?

• The eigenvalues of matrix  are found by solving the characteristic polynomial of the matrix
• For this course, as the characteristic polynomial will always be a quadratic, the polynomial will always generate one of the following:
• two real and distinct eigenvalues,
• one real repeated eigenvalue or
• complex eigenvalues

How do you find the eigenvectors of a matrix?

• A value for  that satisfies the equation is an eigenvector of matrix 
• Any scalar multiple of  will also satisfy the equation and therefore there an infinite number of eigenvectors that correspond to a particular eigenvalue
• STEP 1
  Write 
• STEP 2
  Substitute the eigenvalues into the equation , and form two equations in terms of and 
• STEP 3
  There will be an infinite number of solutions to the equations, so choose one by letting one of the variables be equal to  and using that to find the other variable