On this page, we discuss various types of special matrices, such as diagonal, triangular, and symmetric matrices.
Note: The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to "read ahead" before each class - but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Find the inverse of a diagonal matrix by inspection (or determine that it is singular without any computations).
- Compute matrix products involving diagonal matrices by inspection.
- Determine whether a matrix is triangular.
- Understand how the transpose operation affects diagonal and triangular matrices.
- Understand how the inversion affects diagonal and triangular matrices.
- Determine whether a matrix is a symmetric or skew-symmetric matrix.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Prove results involving diagonal, triangular, symmetric and skew-symmetric matrices.
To prepare for class
Watch this short video which reviews a few special types of matrices: zero, identity, diagonal, banded, and (upper or lower) triangular:
Watch this short video which introduces symmetric and skew-symmetric matrices, and explains the interesting fact that any square matrix can be written as a sum of some symmetric and some skew-symmetric matrix: