IB DP Maths: AI HL復習筆記1.8.1 Eigenvalues & Eigenvectors
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?
How do I find the characteristic polynomial?
Exam Tip
- You need to remember the?characteristic equation?as it is?not?given in the formula booklet
Worked Example
Eigenvalues & Eigenvectors
How do you find the eigenvalues of a 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?
Exam Tip
- You can do a quick check to 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
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