Eigenvalues of a 3x3 matrix | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy

TL;DR

This video explains how to find the eigenvalues of a 3x3 matrix by using the determinant of lambda times the identity matrix minus A.

Transcript

We figured out the eigenvalues for a 2 by 2 matrix, so let's see if we can figure out the eigenvalues for a 3 by 3 matrix. And I think we'll appreciate that it's a good bit more difficult just because the math becomes a little hairier. So lambda is an eigenvalue of A. By definition, if and only if-- I'll write it like this. If and only if A times s... Read More

Key Insights

• 😫 Eigenvalues can be found by setting the determinant of lambda times the identity matrix minus A equal to zero.
• ❓ The characteristic polynomial represents the determinant of the matrix for any lambda.
• 🧑‍🏭 Potential roots of the characteristic polynomial can be found by factoring the constant term of the polynomial.

Questions & Answers

Q: What is the definition of an eigenvalue?

An eigenvalue is a scalar lambda that satisfies the equation A * v = lambda * v, where A is a matrix and v is a non-zero vector.

Q: How can eigenvalues be found for a 3x3 matrix?

Eigenvalues can be found by solving the characteristic equation, which is the determinant of lambda times the identity matrix minus A equal to zero.

Q: Why is it important to find the eigenvalues of a matrix?

Eigenvalues provide important information about a matrix, such as its determinant, trace, and eigenvectors, which can be used to solve systems of linear equations and analyze linear transformations.

Q: How are the possible eigenvalues determined?

The possible eigenvalues are the values of lambda that make the determinant of lambda times the identity matrix minus A equal to zero.

Summary & Key Takeaways

• Eigenvalues for a 3x3 matrix can be found by determining if the determinant of lambda times the identity matrix minus A is equal to zero.

• The characteristic polynomial for a 3x3 matrix is lambda - 3 * (lambda + 3) * (lambda - 3) = 0.

• The possible eigenvalues for the 3x3 matrix are lambda = 3 and lambda = -3.