A quick trick for computing eigenvalues | Chapter 15, Essence of linear algebra

Summary by Krater

This video explains a shortcut method to find the eigenvalues of 2x2 matrices using the trace (mean) and determinant (product), avoiding the characteristic polynomial step.

From the video

Answers: How can I find the eigenvalues of a 2x2 matrix quickly without the characteristic polynomial?

• eigenvalues
• 2x2 matrices
• matrix trace
• matrix determinant
• mean product formula

What it concludes

• The mean of the two eigenvalues of a 2x2 matrix is equal to the mean of its diagonal entries (half the trace).
• The determinant of a 2x2 matrix is equal to the product of its two eigenvalues.
• The two eigenvalues of a 2x2 matrix can be computed directly using the formula m plus or minus the square root of (m squared minus p), where m is the mean of the diagonal entries and p is the determinant.
• Using the mean-product formula avoids calculating the characteristic polynomial and its roots explicitly for 2x2 matrices.