Upper, Lower, and Diagonal Matrices
This page follows the JoeCMath video Upper/Lower Triangular and Diagonal Matrix Explained Fast. The video starts by naming the main diagonal, then uses that diagonal to decide whether a matrix is upper triangular, lower triangular, diagonal, or none of those.
Main Diagonal
The video first points out the main diagonal of a square matrix. In a $3\times 3$ matrix,
\[A= \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix},\]the main diagonal runs from the upper-left entry to the lower-right entry:
\[1,\ 5,\ 9.\]The rest of the lesson uses that diagonal as the dividing line for the entries above and below it.
Watch this section: main diagonal definitions at 0:00.
Upper Triangular Matrices
An upper triangular matrix has zeros below the main diagonal. The entries on the diagonal and above the diagonal can be nonzero.
For example, the video shows the shape
\[\begin{bmatrix} 1 & 5 & 6 \\ 0 & 2 & 7 \\ 0 & 0 & 3 \end{bmatrix}.\]The important part is not that every entry above the diagonal is nonzero. The important part is that every entry below the diagonal is zero.
Watch this section: upper triangular matrix definitions at 0:19.
Lower Triangular Matrices
A lower triangular matrix has zeros above the main diagonal. The entries on the diagonal and below the diagonal can be nonzero.
The matching lower triangular shape in the video is
\[\begin{bmatrix} 1 & 0 & 0 \\ 5 & 2 & 0 \\ 6 & 7 & 3 \end{bmatrix}.\]This time the nonzero entries live below the diagonal instead of above it.
Watch this section: lower triangular matrix definitions at 0:37.
Diagonal Matrices
A diagonal matrix is stricter. Every entry off the main diagonal must be zero.
The video shows a matrix like
\[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}.\]Because all the off-diagonal entries are zero, this matrix is diagonal. It is also upper triangular and lower triangular at the same time.
Watch this section: diagonal matrix definitions at 1:21.
Walkthrough Example
The video then classifies a larger matrix:
\[\begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 2 & 2 & 3 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 5 \end{bmatrix}.\]There are zeros below the main diagonal, so it is upper triangular.
It is not lower triangular because there are nonzero entries above the diagonal, such as the $1$ in the first row and third column.
It is not diagonal because a diagonal matrix cannot have nonzero entries off the main diagonal.
Watch this section: walkthrough classifying example matrix at 1:40.
Practice Matrices
Near the end, the video asks you to classify four matrices. Written out, the examples are
\[A= \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 6 \end{bmatrix}\] \[B= \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 0 & 0 & 6 & 0 \\ 2 & 0 & 0 & 5 \end{bmatrix}\] \[C= \begin{bmatrix} 1 & 0 & 0 \\ 3 & 2 & 0 \\ 1 & 0 & 6 \end{bmatrix}\] \[D= \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}.\]| Matrix | Upper triangular | Lower triangular | Diagonal |
|---|---|---|---|
| $A$ | Yes | No | No |
| $B$ | No | No | No |
| $C$ | No | Yes | No |
| $D$ | Yes | Yes | Yes |
Matrix $A$ has zeros below the main diagonal, so it is upper triangular. Matrix $C$ has zeros above the main diagonal, so it is lower triangular. Matrix $D$ only has nonzero entries on the main diagonal, so it is diagonal, upper triangular, and lower triangular.
Watch this section: practice classifying four examples at 2:14.
Null Matrix
The final question in the video asks what happens with the null matrix:
\[\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}.\]Since every off-diagonal entry is zero, the null matrix is diagonal. It also satisfies the upper triangular and lower triangular conditions because all entries above and below the main diagonal are zero.
Watch this section: is the null matrix upper, lower, or diagonal at 3:34.
Timestamp Guide
Related Linear Algebra Work
After recognizing special matrix shapes, it is natural to keep building matrix vocabulary: