Upper Triangular Matrices
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- Опубликовано: 12 апр 2025
- Every operator on a finite-dimensional complex vector space has an upper-triangular matrix with respect to some basis. The eigenvalues of the operator are the numbers along the diagonal of this upper-triangular matrix.
Your textbook and videos have been a huge help! Been making me fall in love with Linear Algebra!
Great content, was not expecting the upswing in dramatic tension at 5:00 lol
at 2:58, I believe you misspoke. You said this is a "diagonal matrix," I believe you mean to say "upper traingular matrix." Thanks for these videos
7:00 isn't also related to the fundamental theorem of algebra applied to the characteristic polinomial?
The proof given in the book is much simpler than using the characteristic polynomial because the determinant has not yet even been defined by this point in the book.
If M(T) is upper triangular, and assume the diagonal elements are repeated, can T have more eigenvalue other than the diagonal elements? b/c perhaps M(T) wrp to other basis could also be upper triangular.
T cannot have any additional eigenvalues other than the numbers on the diagonal of an upper-triangular matrix with respect to a basis. If the basis changes to another basis with respect to which the matrix is also upper-triangular, then the set of numbers on the diagonal does not change. See Theorem 5.32 in the book.
@@sheldonaxler5197 I got it , thank you professor.
Should the title be Upper Triangular Matrices?
Thank you. I have fixed that typo.
What the hell is the point of you making these videos if you're just going to read verbatim from the book
The videos are not just being read verbatim from the book. You are free not to use the videos if you do not like them. However, different students have different learning styles, and many students find the videos useful, as demonstrated by the over four million minutes that the videos have been viewed. There is no excuse for being nasty about this issue.
We are using your book in the course I'm teaching and students find the videos quite helpful as summaries / alternative media forms. Thanks for recording them!