Chapter 12 Square Matrices and Friends
In this chapter, you will learn about square matrices and several specific types of square matrices encountered in statistical and psychometric applications. Many of these matrices have properties that make them incredibly useful for computation.
12.1 Square Matrices
When the number of rows and columns in a matrix are equal, the matrix is referred to as a square matrix. For example, X and Y are both square matrices.
\[ \underset{2\times 2}{\mathbf{X}} = \begin{bmatrix} 4 & 1 \\ 0 & 3 \end{bmatrix} \qquad \underset{3\times 3}{\mathbf{Y}} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 3 & -8\\ 10 & 4 & -2 \end{bmatrix} \]
12.1.1 Main Diagonal
In a square matrix, the main diagonal (a.k.a., major diagonal or principal diagonal) includes the elements that lie along the diagonal from the upper-left element to the lower-right element. For example, the main diagonal in Y (highlighted in red) includes the elements \(0\), \(3\), and \(-2\). Elements not in the main diagonal are referred to as off-diagonal elements.
\[ \underset{3\times 3}{\mathbf{Y}} = \begin{bmatrix} \color{red}{0} & 1 & 0 \\ 1 & \color{red}{3} & -8\\ 10 & 4 & \color{red}{-2} \end{bmatrix} \]
One interesting and useful property of a square matrix is that if we compute its transpose, the main diagonal is the same as in the original matrix.
\[ \underset{3\times 3}{\mathbf{Y}} = \begin{bmatrix} \color{red}0 & 1 & 0 \\ 1 & \color{red}3 & -8\\ 10 & 4 & \color{red}{-2} \end{bmatrix} \qquad \underset{3\times 3}{\mathbf{Y}^\intercal} = \begin{bmatrix} \color{red}0 & 1 & 10 \\ 1 & \color{red}3 & 4\\ 0 & -8 & \color{red}{-2} \end{bmatrix} \]
We can use the diag() function to return the elements on the main diagonal in a square matrix.
# Create Y
Y = matrix(
data = c(0, 1, 10, 1, 3, 4, 0, -8, -2),
nrow = 3
)
# Display Y
Y [,1] [,2] [,3]
[1,] 0 1 0
[2,] 1 3 -8
[3,] 10 4 -2# Find diagonal elements
diag(Y)[1] 0 3 -2🚧 CAUTION: The diag() function also works on non-square matrices. However, it returns the elements on the diagonal starting with the element in the first row and column.
# Create X
X = matrix(
data = c(0, 1, 1, 3, 4, 0),
nrow = 3
)
# Display X
X [,1] [,2]
[1,] 0 3
[2,] 1 4
[3,] 1 0# Find 'diagonal' elements
diag(X)[1] 0 4This is because non-square matrices also have a main diagonal. The main diagonal in a non-square matrix includes the elements \(a_{11}, a_{22}, \ldots, a_{mm}\) where m is the minimum of the row and column values.
12.2 Diagonal Matrices
A diagonal matrix is a square matrix in which all the off-diagonal elements are zero. Two examples of diagonal matrices are as follows:
\[ \underset{2\times 2}{\mathbf{D}_1} = \begin{bmatrix} 5 & 0 \\ 0 & -1 \end{bmatrix} \qquad \underset{3\times 3}{\mathbf{D}_2} = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} \]
In general, D is a diagonal matrix if element \(d_{ij}=0\) for all \(i\neq j\).
12.3 Scalar Matrices
A diagonal matrix in which all the diagonal elements have the same value is called a scalar matrix. The following two matrices are scalar matrices:
\[ \underset{2\times 2}{\mathbf{S}_1} = \begin{bmatrix} -3 & 0 \\ 0 & -3 \end{bmatrix} \qquad \underset{3\times 3}{\mathbf{S}_2} = \begin{bmatrix} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{bmatrix} \]
The null matrix is also a scalar matrix. In general S is a scalar matrix if element
\[ s_{ij}=\begin{cases}k \quad \mathrm{when} \quad i=j \\0 \quad \mathrm{when} \quad i\neq j\end{cases} \]
12.3.1 Identity Matrices
A special scalar matrix, the identity matrix, is one in which all the diagonal elements are the value of one (1).
\[ \underset{2\times 2}{\mathbf{I}} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \underset{3\times 3}{\mathbf{I}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
The identity matrix has the following property:
- For any matrix A, \(\mathbf{AI} = \mathbf{IA} = \mathbf{A}\)
The diag() function can be used to create an identity matrix. We give this function an argument which provides the number of rows and columns.
# Create a 3x3 identity matrix
I = diag(3)
# Display I
I [,1] [,2] [,3]
[1,] 1 0 0
[2,] 0 1 0
[3,] 0 0 112.4 Symmetric Matrices
If \(\mathbf{A} = \mathbf{A}^{\intercal}\), the matrix A is considered symmetric. For example, the following matrix R is symmetric:
\[ \underset{3\times 3}{\mathbf{R}} = \begin{bmatrix} 1.00 & 0.32 & 0.69\\ 0.32 & 1.00 & 0.85 \\ 0.69 & 0.85 & 1.00 \end{bmatrix} = \underset{3\times 3}{\mathbf{R}^\intercal} = \begin{bmatrix} 1.00 & 0.32 & 0.69\\ 0.32 & 1.00 & 0.85 \\ 0.69 & 0.85 & 1.00 \end{bmatrix} \]
In general, a matrix A is symmetric if,
\[ \mathbf{A}_{ij} = \mathbf{A}_{ji} \]
If matrix A is symmetric, it necessitates that:
- \(\mathbf{A}\) and \(\mathbf{A}^{\intercal}\) have the same dimensions. This also implies that the matrix must be a square matrix.
- All of the corresponding elements in \(\mathbf{A}\) and \(\mathbf{A}^{\intercal}\) are equal.
All scalar matrices, including the identity matrix, are symmetric.
In statistical practice, variance–covariance matrices and correlation matrices are symmetric. Computationally, we can examine whether a matrix is symmetric by checking whether the logical statement equating a matrix and its transpose evaluates as TRUE for all elements. For example, consider the correlation matrix R:
\[ \mathbf{R}=\begin{bmatrix}1.00 & 0.32 & 0.69\\ 0.32 & 1.00 & 0.85\\ 0.69 & 0.85 & 1.00\end{bmatrix} \]
To check its symmetry, we can use the following syntax.
# Create matrix R
R = matrix(
data = c(1.00, 0.32, 0.69, 0.32, 1.00, 0.85, 0.69, 0.85, 1.00),
nrow = 3
)
# Display R
R [,1] [,2] [,3]
[1,] 1.00 0.32 0.69
[2,] 0.32 1.00 0.85
[3,] 0.69 0.85 1.00# Test for symmetry
R == t(R) [,1] [,2] [,3]
[1,] TRUE TRUE TRUE
[2,] TRUE TRUE TRUE
[3,] TRUE TRUE TRUEIf the logical statement of equality evaluates as TRUE for all the elements, the matrix is symmetric. If the logical expression returns an non-conformable array error, or evaluates as FALSE for any of the elements, then it is not symmetric.
12.4.1 Skew Symmetric Matrices
A matrix A is said to be skew symmetric if \(\mathbf{A} = -\mathbf{A}^{\intercal}\).For example, the following matrix S is skew symmetric:
\[ \underset{3\times 3}{\mathbf{S}} = \begin{bmatrix} 0 & 2 & -3\\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix} \]
In general, a matrix A is skew symmetric if, \(\mathbf{a}_{ij} = -\mathbf{a}_{ji}\). Note that if a matrix is skew symmetric, all of its diagonal elements must be zero.
12.5 Triangular Matrices
A matrix is a triangular matrix if all of the elements above or below the main diagonal are zero. These are referred to as upper triangular matrices and lower triangular matrices, respectively. A matrix A is a lower triangular matrix if elements \(a_{ij}=0\) for \(j>i\) (all the elements above the main diagonal are zero). For example the matrix L is lower triangular:
\[ \underset{4\times 4}{\mathbf{L}} = \begin{bmatrix} 6 & \color{red}{0} & \color{red}{0} & \color{red}{0}\\ -1 & 5 & \color{red}{0} & \color{red}{0} \\ 0 & 1 & 2 & \color{red}{0} \\ 3 & -2 & 5 & 7 \end{bmatrix} \]
A matrix A is an upper triangular matrix if elements \(a_{ij}=0\) for \(i>j\) (all the elements below the main diagonal are zero). For example the matrix U is lower triangular:
\[ \underset{4\times 4}{\mathbf{U}} = \begin{bmatrix} 3 & -2 & 5 & 7\\ \color{red}{0} & 1 & 2 & 0 \\ \color{red}{0} & \color{red}{0} & -1 & 5 & \\ \color{red}{0} & \color{red}{0} & \color{red}{0} & 6\\ \end{bmatrix} \]
12.6 Exercises
Consider the following matrices:
\[ \mathbf{A} = \begin{bmatrix}1 & 5 & -1 \\ 3 & 2 & -1 \\ 0 & 1 & 5 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix}3 & 1 & 6 \\ 2 & 0 & 1 \\ -7 & -1 & 2 \end{bmatrix} \]
- List the diagonal elements of \(\mathbf{A}^\intercal\).
- Create an identity matrix of order 4.
- Explain why the transpose of an upper triangular matrix will be a lower triangular matrix.
Consider the following:
\[ \mathbf{X} = \begin{bmatrix} 10 & 4 & 6\\4 & 10 & 8\\6 & 8 & 9 \end{bmatrix} \qquad \mathbf{Y} = \begin{bmatrix}0 & 1 & 2\\-1 & 0 & 4\\-2 & -4 & 0 \end{bmatrix} \]
- Use R to show that X is symmetric.
- Use R to show that Y is skew symmetric.