Characteristic Polynomial - Computation Example
Suppose we want to compute the characteristic polynomial of the matrix
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first, subtract 𝜆𝐼 from 𝐴
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next compute its determinant:
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thus, the characteristic polynomial of 𝐴 is:
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𝜆2 - 2𝜆 + 1
Characteristic Polynomial - Properties
properties of the characteristic polynomial of a 𝑛x𝑛 matrix 𝐴:
- its degree is 𝑛
- is monic (its leading coefficient is 1)
- its roots are the eigenvalues of matrix 𝐴 (proof)
Both matrix 𝐴 and its transpose 𝐴𝑇 have the same characteristic polynomial (proof)