Uploaded October 2024 | Updated September 2026, 1 week ago
A proof that the product of upper triangular matrices is upper triangular. We also prove that the diagonal entries of the matrix product are simply the products of the diagonal entries of the factor matrices.
Learning Linear Algebra playlist: youtube.com/playlist?list=PLug5ZIRrShJHNCfEiX6l5CKbljWayGEcs
0:00 Upper Triangular Definition
1:25 Product is Upper Triangular
6:42 Diagonal Entries
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Music: C418 - Smooth Fall
A proof that the product of upper triangular matrices is upper triangular. We also prove that the diagonal entries of the matrix product are simply the products of the diagonal entries of the factor matrices.
Learning Linear Algebra playlist: youtube.com/playlist?list=PLug5ZIRrShJHNCfEiX6l5CKbljWayGEcs
0:00 Upper Triangular Definition
1:25 Product is Upper Triangular
6:42 Diagonal Entries
Subscribe to see more new math videos!
Music: C418 - Smooth Fall
![F[x]-Module Derivation of Rational and Jordan Canonical Forms](https://i.ytimg.com/vi/q5uj4o0O5R0/mqdefault.jpg "F[x]-Module Derivation of Rational and Jordan Canonical Forms
Similar matrices isomorphism proof: https://youtu.be/-ligAAxFM8Y
Every module is a direct sum of cyclic modules: https://youtu.be/gWIRI43h0ic
Intro to F[x]-modules: https://youtu.be/H44q_Urmts0
The rational canonical form and Jordan normal form of a matrix are very important tools in linear algebra, but ring theory and module theory give us a very effective way to prove their existence! Here we show that every matrix is similar to a matrix in rational and Jordan canonical form.
Ring & Module Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 Rational canonical form
14:17 Every matrix is similar to RCF
18:33 Algebraically closed fields
20:10 Jordan canonical form
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Music: OcularNebula - The Lopez")
