Example solving for the eigenvalues of a 2x2 matrix | Linear Algebra | Khan Academy

TL;DR

Understanding eigenvalues and eigenvectors and how to find them for a given matrix.

Transcript

In the last video we were able to show that any lambda that satisfies this equation for some non-zero vectors, V, then the determinant of lambda times the identity matrix minus A, must be equal to 0. Or if we could rewrite this as saying lambda is an eigenvalue of A if and only if-- I'll write it as if-- the determinant of lambda times the identity... Read More

Key Insights

• ⌛ Eigenvalues of a matrix can be found using the equation lambda times the identity matrix minus A equals 0.
• 💨 The characteristic polynomial is derived from the determinant of the matrix equation and provides a way to solve for eigenvalues.
• ⌛ The determinant of lambda times the identity matrix minus A should be equal to 0 to find eigenvalues.

Questions & Answers

Q: What is an eigenvalue and eigenvector?

Eigenvalues and eigenvectors are important concepts in linear algebra. An eigenvalue represents a scalar value associated with an eigenvector, a vector that remains in the same direction when multiplied by a transformation matrix.

Q: How can eigenvalues be found for a matrix?

Eigenvalues can be found by solving the characteristic polynomial equation, which is derived from the determinant of the matrix equation lambda times the identity matrix minus A equals 0.

Q: What does the determinant of lambda times the identity matrix minus A represent?

The determinant of lambda times the identity matrix minus A represents the determinant of the matrix formed by subtracting the matrix A from a multiple of the identity matrix. This determinant needs to be equal to 0 to find the eigenvalues.

Q: Why does the characteristic polynomial need to be solved to find eigenvalues?

The characteristic polynomial is a polynomial equation that can be factored to find the values of lambda (eigenvalues) that satisfy the equation lambda times the identity matrix minus A equals 0.

Summary & Key Takeaways

• The video introduces the concept of eigenvalues and shows how they can be derived using the determinant of a matrix equation.

• It demonstrates the process of finding eigenvalues for a 2 by 2 matrix by solving a characteristic polynomial equation.

• The video concludes by stating the eigenvalues found and mentioning that the next video will focus on finding eigenvectors.