Finding eigenvectors and eigenspaces example | Linear Algebra | Khan Academy | Summary and Q&A

TL;DR

The video explains what eigenvalues and eigenvectors are and how to find them.

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.

Transcript

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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.