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Eigenvalues & Eigenvectors (DP IB Maths: AI HL)
Revision Note
Characteristic Polynomials
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
Examiner Tip
- You need to remember the characteristic equation as it is not given in the formula booklet
Worked example
Find the characteristic polynomial of the following matrix
.
Eigenvalues & Eigenvectors
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
Examiner Tip
- You can do a quick check on your calculated eigenvalues as the values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix
Worked example
Find the eigenvalues and associated eigenvectors for the following matrices.
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