Fact

Matrix 𝐴 and its transpose 𝐴^𝑇 have the same:

• characteristic polynomial
• eigenvalues

Proof - Same Characteristic Polynomial

The characteristic polynomial of 𝐴 is defined as:

• 𝑑𝑒𝑡𝑒𝑚𝑖𝑛𝑎𝑛𝑡(𝐴 - 𝜆𝐼)

The determinant of a matrix and its transpose are the same (see here). Thus:

• 𝑑𝑒𝑡𝑒𝑚𝑖𝑛𝑎𝑛𝑡(𝐴 - 𝜆𝐼) = 𝑑𝑒𝑡𝑒𝑚𝑖𝑛𝑎𝑛𝑡((𝐴 - 𝜆𝐼)^𝑇)

Simplifying the right-hand side yields:

• 𝑑𝑒𝑡𝑒𝑚𝑖𝑛𝑎𝑛𝑡(𝐴 - 𝜆𝐼) = 𝑑𝑒𝑡𝑒𝑚𝑖𝑛𝑎𝑛𝑡(𝐴^𝑇 - 𝜆𝐼)

The characteristic polynomial defined on the right is of 𝐴^𝑇

Proof - Same Eigenvalues

Since both 𝐴 and 𝐴^𝑇have the same characteristic polynomial. They necessarily have the same eigenvalues.