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.
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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.