Definition 2.1.1. Zero Matrix.
A zero matrix is a matrix in which all entries are zero. We denote an \(m \times n\) zero matrix by \(O_{m \times n}\) or simply \(O\) when the dimensions are clear from context.
Section 2.1 Special Types of Matrices
Before we explore matrix operations, it’s important to become familiar with several special types of matrices that appear frequently in linear algebra and its applications. These special matrices have unique properties that make them particularly useful in various contexts, from solving systems of equations to representing transformations.
Understanding these matrix types will help us recognize patterns, simplify computations, and appreciate the elegant structure that underlies many mathematical operations.
Subsection 2.1.1 Zero and Identity Matrices
Example 2.1.2. Examples of Zero Matrices.
\begin{align*}
O_{2 \times 3} \amp= \begin{pmatrix} 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \end{pmatrix}\\
O_{3 \times 3} \amp= \begin{pmatrix} 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \end{pmatrix}
\end{align*}
The zero matrix will play a similar role in matrix algebra as the number 0 does in ordinary arithmetic.
Definition 2.1.3. Identity Matrix.
An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. The \(n \times n\) identity matrix is denoted by \(I_n\) or simply \(I\text{.}\) Formally, \(I_n = [a_{ij}]\) where
\begin{equation*}
a_{ij} = \begin{cases}
1 \amp \text{if } i = j \\
0 \amp \text{if } i \neq j
\end{cases}
\end{equation*}
Example 2.1.4. Examples of Identity Matrices.
\begin{align*}
I_2 \amp= \begin{pmatrix} 1 \amp 0 \\ 0 \amp 1 \end{pmatrix}\\
I_3 \amp= \begin{pmatrix} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \end{pmatrix}\\
I_4 \amp= \begin{pmatrix} 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \end{pmatrix}
\end{align*}
The identity matrix will play a similar role in matrix multiplication as the number 1 does in ordinary arithmetic.
Subsection 2.1.2 Diagonal and Scalar Matrices
Definition 2.1.5. Diagonal Matrix.
A square matrix \(A = [a_{ij}]\) is called a diagonal matrix if \(a_{ij} = 0\) for all \(i \neq j\text{.}\) In other words, all entries off the main diagonal are zero.
We often write a diagonal matrix as \(\text{diag}(d_1, d_2, \ldots, d_n)\) where \(d_1, d_2, \ldots, d_n\) are the diagonal entries.
Example 2.1.6. Examples of Diagonal Matrices.
\begin{align*}
D_1 \amp= \begin{pmatrix} 3 \amp 0 \amp 0 \\ 0 \amp -2 \amp 0 \\ 0 \amp 0 \amp 5 \end{pmatrix} = \text{diag}(3, -2, 5)\\
D_2 \amp= \begin{pmatrix} 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 4 \amp 0 \amp 0 \\ 0 \amp 0 \amp -1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 2 \end{pmatrix} = \text{diag}(1, 4, -1, 2)
\end{align*}
Diagonal matrices have a simple structure that makes them particularly easy to work with in many linear algebra applications.
Definition 2.1.7. Scalar Matrix.
A scalar matrix is a diagonal matrix in which all diagonal entries are equal. A scalar matrix has the form \(\lambda I_n\) for some scalar \(\lambda\text{.}\)
Example 2.1.8. Examples of Scalar Matrices.
\begin{align*}
3I_2 \amp= \begin{pmatrix} 3 \amp 0 \\ 0 \amp 3 \end{pmatrix}\\
-5I_3 \amp= \begin{pmatrix} -5 \amp 0 \amp 0 \\ 0 \amp -5 \amp 0 \\ 0 \amp 0 \amp -5 \end{pmatrix}
\end{align*}
Scalar matrices are diagonal matrices where all diagonal entries have the same value. They represent uniform scaling in all directions.
Subsection 2.1.3 Triangular Matrices
Definition 2.1.9. Upper Triangular Matrix.
A square matrix \(A = [a_{ij}]\) is called upper triangular if \(a_{ij} = 0\) for all \(i > j\text{.}\) In other words, all entries below the main diagonal are zero.
Definition 2.1.10. Lower Triangular Matrix.
A square matrix \(A = [a_{ij}]\) is called lower triangular if \(a_{ij} = 0\) for all \(i < j\text{.}\) In other words, all entries above the main diagonal are zero.
Example 2.1.11. Examples of Triangular Matrices.
Upper triangular matrix:
\begin{equation*}
U = \begin{pmatrix}
2 \amp 5 \amp -3 \amp 1 \\
0 \amp 4 \amp 7 \amp -2 \\
0 \amp 0 \amp -1 \amp 6 \\
0 \amp 0 \amp 0 \amp 3
\end{pmatrix}
\end{equation*}
Lower triangular matrix:
\begin{equation*}
L = \begin{pmatrix}
3 \amp 0 \amp 0 \amp 0 \\
-2 \amp 1 \amp 0 \amp 0 \\
5 \amp 4 \amp 2 \amp 0 \\
1 \amp -3 \amp 6 \amp -4
\end{pmatrix}
\end{equation*}
Triangular matrices are important in numerical linear algebra because systems of equations involving triangular matrices can be solved efficiently using back-substitution (for upper triangular) or forward-substitution (for lower triangular).
Definition 2.1.12. Strictly Triangular Matrices.
-
A square matrix \(A = [a_{ij}]\) is called strictly upper triangular if \(a_{ij} = 0\) for all \(i \geq j\text{.}\) (All entries on and below the main diagonal are zero.)
-
A square matrix \(A = [a_{ij}]\) is called strictly lower triangular if \(a_{ij} = 0\) for all \(i \leq j\text{.}\) (All entries on and above the main diagonal are zero.)
Example 2.1.13. Examples of Strictly Triangular Matrices.
Strictly upper triangular matrix:
\begin{equation*}
U_s = \begin{pmatrix}
0 \amp 5 \amp -3 \amp 1 \\
0 \amp 0 \amp 7 \amp -2 \\
0 \amp 0 \amp 0 \amp 6 \\
0 \amp 0 \amp 0 \amp 0
\end{pmatrix}
\end{equation*}
Strictly lower triangular matrix:
\begin{equation*}
L_s = \begin{pmatrix}
0 \amp 0 \amp 0 \amp 0 \\
-2 \amp 0 \amp 0 \amp 0 \\
5 \amp 4 \amp 0 \amp 0 \\
1 \amp -3 \amp 6 \amp 0
\end{pmatrix}
\end{equation*}
Note that strictly triangular matrices always have zeros on the main diagonal, which distinguishes them from general triangular matrices.
Subsection 2.1.4 Symmetric Matrices
Definition 2.1.14. Symmetric Matrix.
A square matrix \(A = [a_{ij}]\) is called symmetric if \(a_{ij} = a_{ji}\) for all \(i\) and \(j\text{.}\) In other words, the entries are symmetric with respect to the main diagonal.
Equivalently, a matrix is symmetric if it equals its transpose: \(A = A^T\text{,}\) where the transpose \(A^T\) is obtained by reflecting \(A\) across its main diagonal (swapping rows and columns).
Example 2.1.15. Examples of Symmetric Matrices.
A \(2 \times 2\) symmetric matrix:
\begin{equation*}
A_1 = \begin{pmatrix}
3 \amp 5 \\
5 \amp -2
\end{pmatrix}
\end{equation*}
Note that the entry in position \((1,2)\) equals the entry in position \((2,1)\text{.}\)
A \(3 \times 3\) symmetric matrix:
\begin{equation*}
A_2 = \begin{pmatrix}
4 \amp 1 \amp -3 \\
1 \amp 0 \amp 7 \\
-3 \amp 7 \amp 2
\end{pmatrix}
\end{equation*}
Observe that \(a_{12} = a_{21} = 1\text{,}\) \(a_{13} = a_{31} = -3\text{,}\) and \(a_{23} = a_{32} = 7\text{.}\)
A \(4 \times 4\) symmetric matrix:
\begin{equation*}
A_3 = \begin{pmatrix}
2 \amp -1 \amp 0 \amp 4 \\
-1 \amp 5 \amp 3 \amp 1 \\
0 \amp 3 \amp -2 \amp 6 \\
4 \amp 1 \amp 6 \amp 0
\end{pmatrix}
\end{equation*}
Check that each entry above the diagonal has a matching entry below the diagonal.
Example 2.1.16. Special Cases of Symmetric Matrices.
Several of the special matrix types we’ve already seen are also symmetric:
-
All diagonal matrices are symmetric: Since \(a_{ij} = 0\) for \(i \neq j\text{,}\) we automatically have \(a_{ij} = a_{ji}\text{.}\)\begin{equation*} \begin{pmatrix} 5 \amp 0 \amp 0 \\ 0 \amp -3 \amp 0 \\ 0 \amp 0 \amp 7 \end{pmatrix} \end{equation*}
-
The identity matrix is symmetric:\begin{equation*} I_3 = \begin{pmatrix} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \end{pmatrix} \end{equation*}
-
The zero matrix is symmetric:\begin{equation*} O_{3\times 3} = \begin{pmatrix} 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \end{pmatrix} \end{equation*}
Example 2.1.17. Non-Symmetric Matrices.
This matrix is NOT symmetric:
\begin{equation*}
B = \begin{pmatrix}
1 \amp 2 \amp 3 \\
4 \amp 5 \amp 6 \\
7 \amp 8 \amp 9
\end{pmatrix}
\end{equation*}
For example, \(b_{12} = 2\) but \(b_{21} = 4\text{,}\) so \(b_{12} \neq b_{21}\text{.}\)
Triangular matrices (except diagonal ones) are generally not symmetric:
\begin{equation*}
U = \begin{pmatrix}
1 \amp 2 \amp 3 \\
0 \amp 4 \amp 5 \\
0 \amp 0 \amp 6
\end{pmatrix}
\end{equation*}
Here \(u_{12} = 2\) but \(u_{21} = 0\text{,}\) so this upper triangular matrix is not symmetric.
Remark 2.1.18. Importance of Symmetric Matrices.
Symmetric matrices arise naturally in many applications:
-
Distance matrices in geometry (distance from point \(i\) to point \(j\) equals distance from \(j\) to \(i\))
-
Correlation and covariance matrices in statistics
-
Adjacency matrices of undirected graphs
-
Matrices representing quadratic forms in optimization
Symmetric matrices have many special properties that we will explore later in the course, including the fact that they always have real eigenvalues and can be diagonalized using orthogonal matrices.
Subsection 2.1.5 Summary and Relationships
The following diagram illustrates the relationships between these special matrix types:
Insight 2.1.19. Hierarchy of Special Matrices.
-
Every scalar matrix is a diagonal matrix.
-
Every diagonal matrix is both upper triangular and lower triangular.
-
Every diagonal matrix is symmetric.
-
The identity matrix \(I_n\) is a scalar matrix (with \(\lambda = 1\)), and therefore also diagonal and symmetric.
-
The zero matrix is symmetric, and is also both strictly upper triangular and strictly lower triangular.
-
Strictly triangular matrices are special cases of triangular matrices (with zero diagonal).
-
A symmetric matrix that is also triangular must be diagonal.
Activity 2.1.1. Classifying Matrices.
For each of the following matrices, identify all applicable special types (zero, identity, scalar, diagonal, upper triangular, lower triangular, strictly upper triangular, strictly lower triangular, symmetric):
-
\(\displaystyle A = \begin{pmatrix} 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \end{pmatrix}\)
-
\(\displaystyle B = \begin{pmatrix} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \end{pmatrix}\)
-
\(\displaystyle C = \begin{pmatrix} 2 \amp 0 \amp 0 \\ 0 \amp 2 \amp 0 \\ 0 \amp 0 \amp 2 \end{pmatrix}\)
-
\(\displaystyle D = \begin{pmatrix} 3 \amp 0 \amp 0 \\ 0 \amp -1 \amp 0 \\ 0 \amp 0 \amp 4 \end{pmatrix}\)
-
\(\displaystyle E = \begin{pmatrix} 1 \amp 2 \amp 3 \\ 0 \amp 4 \amp 5 \\ 0 \amp 0 \amp 6 \end{pmatrix}\)
-
\(\displaystyle F = \begin{pmatrix} 0 \amp 2 \amp 3 \\ 0 \amp 0 \amp 5 \\ 0 \amp 0 \amp 0 \end{pmatrix}\)
-
\(\displaystyle G = \begin{pmatrix} 5 \amp 1 \amp -2 \\ 1 \amp 3 \amp 4 \\ -2 \amp 4 \amp 0 \end{pmatrix}\)
-
\(\displaystyle H = \begin{pmatrix} 2 \amp 0 \amp 0 \\ 1 \amp 3 \amp 0 \\ 4 \amp 5 \amp 6 \end{pmatrix}\)
Hint.
-
Check if all entries are zero.
-
Check if there are 1’s on the diagonal and 0’s elsewhere.
-
Check if all diagonal entries are equal.
-
Check if \(a_{ij} = 0\) for \(i \neq j\) (only diagonal entries can be non-zero).
-
Check if \(a_{ij} = 0\) for \(i > j\) (entries below diagonal are zero).
-
Check if \(a_{ij} = 0\) for \(i < j\) (entries above diagonal are zero).
-
Check if \(a_{ij} = a_{ji}\) for all \(i, j\) (symmetric about diagonal).
