IB DP Maths: AI HL复习笔记1.8.2 Applications of Matrices
Diagonalisation
What is matrix diagonalisation?
- A non-zero, square matrix is considered to be diagonal if all elements not along its leading diagonal are zero
You will only need to consider matrices with real, distinct eigenvalues
- If there is only one eigenvalue, the matrix is either already diagonalised or cannot be diagonalised
- Diagonalisation of matrices with complex or imaginary eigenvalues is outside the scope of the course
Exam Tip
- Remember to use the formula booklet for the determinant and inverse of a matrix
Worked Example
Matrix Powers
One of the main applications of diagonalising a matrix is to make it easy to find powers of the matrix, which is useful when modelling transient situations such as the movement of populations between two towns.
How can the diagonalised matrix be used to find higher powers of the original matrix?
Exam Tip
- If you are asked to show this by hand, don’t forget to use your GDC to check your answer afterwards!
Worked Example
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