Advanced Linear Algebra Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors
Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors Transcript and Lesson Notes
Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors Throughout this lecture and the others in this section, we will assume that X is a vector space over an algebraically closed field K, like the complex nu
Quick Summary
Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors Throughout this lecture and the others in this section, we will assume that X is a vector space over an algebraically closed field K, like the complex nu
Key Takeaways
- Review the core idea: Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors Throughout this lecture and the others in this section, we will assume that X is a vector space over an algebraically closed field K, like the complex nu
- Understand how Linear algebra fits into Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors.
- Understand how matrix analysis fits into Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors.
- Understand how eigenvalue fits into Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors.
- Understand how eigenvector fits into Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors.
Key Concepts
Full Transcript
Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors Throughout this lecture and the others in this section, we will assume that X is a vector space over an algebraically closed field K, like the complex numbers. This ensures that every polynomial in K[x] has a root in K. An eigenvector of a linear map A:X→X is any nonzero v such that Av=λv, for some scalar λ called an eigenvalue. Equivalently, v is in the null space of A-λI, and so one characterization of an eigenvalue is any scalar for which det(A-λI)=0. This is how we actually find eigenvalues, and then we find the eigenvector by solving (A-λI)v=0. We show why every linear map has an eigenvector, and why eigenvectors corresponding to distinct eigenvalues are linearly independent. If X has a basis of eigenvectors of A, then we say that A is diagonalizable, because the matrix with respect to this basis is diagonal. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html
Lesson FAQs
What is Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors about?
Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors Throughout this lecture and the others in this section, we will assume that X is a vector space over an algebraically closed field K, like the complex nu
What key concepts are covered in this lesson?
The lesson covers Linear algebra, matrix analysis, eigenvalue, eigenvector, linear map.
What should I learn before Advanced Linear Algebra, Lecture 4.1: Eigenvalues and eigenvectors?
Review the previous lessons in Advanced Linear Algebra, then use the transcript and key concepts on this page to fill any gaps.
How can I practice after this lesson?
Practice by applying the main concepts: Linear algebra, matrix analysis, eigenvalue, eigenvector.
Does this lesson include a transcript?
Yes. The full transcript is visible on this page in indexable HTML sections.
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