Theorem

Eigenvalues are the roots of the Characteristic Polynomial

Let:

• 𝐴 be an 𝑛⨯𝑛 matrix
• 𝑓(𝜆) = 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡(𝐴 - 𝜆𝐼) be its characteristic polynomial

Then a number 𝜆₀ is an eigenvalue of 𝐴 ↔ 𝑓(𝜆₀) = 0

Proof

By the invertible matrix theorem, the matrix equation (𝐴 - 𝜆𝐼)𝑥 = 0 has a non-trivial solution if and only if 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡(𝐴 - 𝜆𝐼) = 0. Therefore:

• 𝜆₀ is an eigenvalue of 𝐴 ↔ 𝐴𝑥 = 𝜆₀𝑥 has a non-trivial solution
• 𝜆₀ is an eigenvalue of 𝐴 ↔ (𝐴 - 𝜆₀𝐼)𝑥 = 0 has a non-trivial solution
• 𝜆₀ is an eigenvalue of 𝐴 ↔ (𝐴 - 𝜆₀𝐼) is not invertible
• 𝜆₀ is an eigenvalue of 𝐴 ↔ 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡(𝐴 - 𝜆₀𝐼) = 0
• 𝜆₀ is an eigenvalue of 𝐴 ↔ 𝑓(𝜆₀) = 0