In this lesson, we learn how to use transposes of matrices and inverses, as well as permutation matrices, to factorize a matrix into A = LU. This factorization is a great way to look at Gaussian elimination, which is the process of eliminating variables in a system of linear equations to get to row echelon form. We also explore how to find the inverse of a product of matrices, and how to separate out the pivots in the matrix using a diagonal matrix. Overall, this lesson provides a solid foundation for understanding matrix factorization and Gaussian elimination.
Factorization into A = LU -- Lecture 4. How to use transposes of matrices and inverses, and permutation matrices.
Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 16, 2008). License: Creative Commons BY-NC-SA.
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Questions answered by this video:
When is a matrix invertible?
What is the transpose of a matrix?
What is LU?
How are permutations used in Linear Algebra?
What is a permutation matrix?
When is the inverse of a matrix just its transpose?
Currently 3.0/5 Stars.
This lesson elaborates on lecture #3 and delves a bit deeper into transposes and does a good job of introducing permutation matrices. A good but somewhat redundant lecture.