How to Find Eigenvectors and Eigenspaces in Linear Algebra

TL;DR

To find eigenvectors and eigenspaces, start by determining the eigenvalues using the characteristic polynomial, defined by the determinant of (lambda * identity - A) = 0. Then, solve the equation (lambda * identity - A) * v = 0 for each eigenvalue to obtain the corresponding eigenvectors, with the eigenspace being the set of all such vectors.

Transcript

In the last video, we started with the 2 by 2 matrix A is equal to 1, 2, 4, 3. And we used the fact that lambda is an eigenvalue of A, if and only if, the determinate of lambda times the identity matrix-- in this case it's a 2 by 2 identity matrix-- minus A is equal to 0. This gave us a characteristic polynomial and we solved for that and we said, ... Read More

Key Insights

• 😫 Eigenvalues are found by solving the characteristic polynomial, which is obtained by setting the determinant of (lambda * identity - A) equal to zero.
• 🥰 Eigenvectors are the vectors that satisfy the equation (lambda * identity - A) * v = 0, where lambda is an eigenvalue and A is the matrix.
• 📼 The eigenspace for a specific eigenvalue is the set of all eigenvectors that satisfy (lambda * identity - A) * v = 0.
• 🫥 The eigenspaces for different eigenvalues can be represented as lines or spans in R2 or higher-dimensional spaces.

Questions & Answers

Q: What is an eigenvalue?

An eigenvalue is a scalar that represents the scaling factor of an eigenvector when it is transformed by a matrix.

Q: How do you find eigenvectors?

To find eigenvectors, solve the equation (lambda * identity - A) * v = 0, where lambda is an eigenvalue and A is the matrix.

Q: What is the eigenspace?

The eigenspace is the set of all eigenvectors that correspond to a specific eigenvalue, and it can be found by calculating the null space of (lambda * identity - A).

Q: How do you determine the eigenvalues of a matrix?

The eigenvalues of a matrix can be found by solving the characteristic polynomial, which is obtained by setting the determinant of (lambda * identity - A) equal to zero.

Summary & Key Takeaways

• Eigenvalues and eigenvectors are important concepts in linear algebra.

• Eigenvalues are scalars that represent the scaling factor of the eigenvectors when transformed by a matrix.

• Eigenvectors are vectors that remain in the same direction but are scaled by the eigenvalue when transformed by a matrix.