.
M5 Special matrices
August 11, 2022 - 5:04pm — AJ (e67821)
It is helpful to understand the definition of a number of different types of “special” matrices.
Transpose of a matrix
The transpose of a matrix \(\mathbf{A}\) is denoted \(\mathbf{A}^{T}\)and is found by interchanging the rows and the columns.
The first row becomes the first column, the second row becomes the second column etc.
If \(\mathbf{A}\) is an \(m\times n\) matrix, then \(\mathbf{A}^{T}\) is an \(n\times m\) matrix.
Examples
- If \[\begin{align*} \mathbf{A} & =\left[\begin{array}{cc} 1 & 2\\ 3 & 4\\ 5 & 6 \end{array}\right] \end{align*}\] then by interchanging the rows and columns we get \[\begin{align*} \mathbf{A}^{T} & =\left[\begin{array}{ccc} 1 & 3 & 5\\ 2 & 4 & 6 \end{array}\right]. \end{align*}\]
In this case \(\mathbf{A}\) is a \(3\times2\) matrix and its transpose \(\mathbf{A}^{T}\) is a \(2\times3\) matrix.
If \[\begin{align*} \mathbf{B} & =\left[\begin{array}{cc} 1 & 2\\ 7 & 5 \end{array}\right] \end{align*}\] then \[\begin{align*} \mathbf{B}^{T} & =\left[\begin{array}{cc} 1 & 7\\ 2 & 5 \end{array}\right]. \end{align*}\]
If \[\begin{align*} \boldsymbol{\mathbf{C}} & =\left[\begin{array}{cc} 5 & 2\\ 2 & 4 \end{array}\right] \end{align*}\] then \[\begin{align*} \mathbf{C}^{T} & =\left[\begin{array}{cc} 5 & 2\\ 2 & 4 \end{array}\right]. \end{align*}\]
Note that in this case \(\mathbf{C}=\mathbf{C}^{T}\) and \(\mathbf{C}\) is called a symmetric matrix.
Symmetric Matrix
A symmetric matrix is a square matrix which is equal to its transpose. It is also symmetric about its leading diagonal (top left to bottom right).
Examples
- The matrix
\[\begin{align*} \mathbf{D} & =\left[\begin{array}{cc} 1 & 3\\ 3 & 4 \end{array}\right] \end{align*}\] is symmetric because \[\begin{align*} \mathbf{D}^{T} & =\left[\begin{array}{cc} 1 & 3\\ 3 & 4 \end{array}\right]\\ & =\mathbf{D}. \end{align*}\] 2. The matrix
\[\begin{align*} \mathbf{E} & =\left[\begin{array}{ccc} 3 & 4 & 5\\ 4 & -2 & -3\\ 5 & -3 & 1 \end{array}\right] \end{align*}\] is symmetric because \[\begin{align*} \mathbf{E}^{T} & =\left[\begin{array}{ccc} 3 & 4 & 5\\ 4 & -2 & -3\\ 5 & -3 & 1 \end{array}\right]\\ & =\mathbf{E}. \end{align*}\] 3. The matrix \[\begin{align*} \mathbf{F} & =\left[\begin{array}{ccc} 1 & -2 & 4\\ -2 & 3 & -4\\ 4 & -4 & 2 \end{array}\right] \end{align*}\] is symmetric because \[\begin{align*} \mathbf{F}^{T} & =\left[\begin{array}{ccc} 1 & -2 & 4\\ -2 & 3 & -4\\ 4 & -4 & 2 \end{array}\right]\\ & =\mathbf{F}. \end{align*}\]
Orthogonal Matrix
A square matrix is orthogonal if \(\mathbf{A}^{T}\mathbf{A}=\mathbf{A}\mathbf{A}^{T}=\mathbf{I}\) where \(\mathbf{I}\) is the unit matrix (also called the identity matrix).
Because \(\mathbf{A}^{-1}\mathbf{A}=\boldsymbol{\mathbf{A}}\mathbf{A}^{-1}=\mathbf{I}\) it follows that for an orthogonal matrix \(\mathbf{A}^{T}=\mathbf{A}^{-1}\) . This can be useful for finding the inverse of an orthogonal matrix as it is usually easier to find the transpose than the inverse of a matrix.
Note that the determinant of an orthhogonal matrix is either equal to \(+1\) (a rotation matrix) or \(-1\) (a reflection matrix).
The rows of an orthogonal matrix are mutually orthogonal (perpendicular) unit vectors. The columns of an orthogonal matrix are also mutually orthogonal unit vectors.
Examples
The rotation matrix \(\mathbf{A}=\left[\begin{array}{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{array}\right]\) is an orthogonal matrix that rotates points, lines and regions through an angle of \(\theta^{\circ}\) anticlockwise. 1 Note that in the following sections, we use the trigonometric identity \[\begin{align*} \cos^{2}\theta+\sin^{2}\theta & =1. \end{align*}\]
The determinant, \(\det\left|\mathbf{A}\right|=\left|\begin{array}{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{array}\right|=\cos^{2}\theta+\sin^{2}\theta=1.\)
\(\mathbf{A}^{T}=\left[\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right]\)
\[\begin{align*} \mathbf{A}^{T}\mathbf{A} & =\left[\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right]\left[\begin{array}{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{array}\right]\\ & =\left[\begin{array}{cc} \cos^{2}\theta+\sin^{2}\theta & \cos\theta\sin\theta-\sin\theta\cos\theta\\ \sin\theta\cos\theta-\cos\theta\sin\theta & \sin^{2}\theta+\cos^{2}\theta \end{array}\right]\\ & =\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right] \end{align*}\] and \[\begin{align*} \mathbf{A}\mathbf{A}^{T} & =\left[\begin{array}{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{array}\right]\left[\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right]\\ & =\left[\begin{array}{cc} \cos^{2}\theta+\sin^{2}\theta & -\cos\theta\sin\theta+\sin\theta\cos\theta\\ -\sin\theta\cos\theta+\cos\theta\sin\theta & \sin^{2}\theta+\cos^{2}\theta \end{array}\right]\\ & =\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]. \end{align*}\]
That is to say \(\mathbf{A}^{T}\mathbf{A}=\mathbf{A}\mathbf{A}^{T}=\mathbf{I}\). Hence
\[\begin{align*} \mathbf{A}^{-1} & =\frac{1}{\det\mathbf{A}}\left[\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right]\\ & =\frac{1}{1}\left[\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right]\\ & =\mathbf{A}^{T}. \end{align*}\]
The reflection matrix \(\mathbf{B}=\left[\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right]\) is an orthogonal matrix that reflects points, lines and regions in the \(x\)-axis.
\(\left|\mathbf{B}\right|=-1\) and \(\mathbf{B}^{T}=\left[\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right]\)
\(\mathbf{B}^{T}\mathbf{B}=\mathbf{B}\mathbf{B}^{T}=\mathbf{I}\) and \(\mathbf{B}^{T}=\mathbf{B}^{-1}\)
Note that in this case \(\mathbf{B}\) is both orthogonal and symmetric.
Exercise 1
Given \[\begin{align*} \mathbf{A} & =\left[\begin{array}{cc} 3 & -2\end{array}\right]\quad\mathbf{B}=\left[\begin{array}{cc} 1 & -3\\ -3 & 0 \end{array}\right]\quad\mathbf{C}=\left[\begin{array}{ccc} 2 & 0 & 5\\ 0 & -2 & 2\\ 5 & 2 & 1 \end{array}\right]\quad\mathbf{D=\left[\mathrm{\begin{array}{ccc} 2 & -1 & 5\\ 5 & -1 & 2 \end{array}}\right]\quad E=\left[\mathrm{\begin{array}{ccc} 0 & 1 & -1\\ 1 & 0 & 1\\ 1 & -1 & 0 \end{array}}\right]}, \end{align*}\]
Write down the transpose of each of the matrices.
Which of the given matrices are symmetric?
\[\begin{align*} \text{a. }\mathbf{A}^{T} & =\left[\begin{array}{c} 3\\ -2 \end{array}\right]\quad\mathbf{B}^{T}=\left[\begin{array}{cc} 1 & -3\\ -3 & 0 \end{array}\right]\quad\mathbf{C}^{T}=\left[\begin{array}{ccc} 2 & 0 & 5\\ 0 & -2 & 2\\ 5 & 2 & 1 \end{array}\right]\quad\mathbf{D}^{T}=\left[\begin{array}{cc} 2 & 5\\ -1 & -1\\ 5 & 2 \end{array}\right]\quad\mathbf{E}=\left[\begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & -1\\ -1 & 1 & 0 \end{array}\right] \end{align*}\]
\(\text{$\text{b.}$ }\mathbf{B}\) and \(\mathbf{C}\) are symmetric.
Exercise 2
Given \[\begin{align*} \mathbf{L} & =\left[\begin{array}{cc} 1 & -1\\ -1 & 0 \end{array}\right]\ \ \mathbf{M}=\left[\begin{array}{ccc} \frac{3}{5} & -\frac{4}{5} & 0\\ \frac{4}{5} & \frac{3}{5} & 0\\ 0 & 0 & 1 \end{array}\right]\ \ \mathbf{N}=\dfrac{1}{3}\left[\begin{array}{ccc} 2 & -2 & 1\\ 1 & 2 & 2\\ 2 & 1 & -2 \end{array}\right]\ \ \mathbf{P}=\left[\begin{array}{ccc} 0 & 1 & -1\\ 1 & 0 & 1\\ 1 & -1 & 0 \end{array}\right]\ \ \mathbf{Q}=\left[\begin{array}{ccc} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & -1 \end{array}\right], \end{align*}\]
Which of the given matrices are orthogonal? If the matrix is orthogonal, write down its inverse.
Which of the given matices is a rotation matrix?
- \(\mathbf{M}\), \(\mathbf{N}\) and \(\mathbf{Q}\) are orthogonal
\[\begin{align*} \mathbf{\qquad M}^{-1} & =\left[\begin{array}{ccc} \frac{3}{5} & \frac{4}{5} & 0\\ -\frac{4}{5} & \frac{3}{5} & 0\\ 0 & 0 & 1 \end{array}\right]\ \ \mathbf{N}^{-1}=\dfrac{1}{3}\left[\begin{array}{ccc} 2 & 1 & 2\\ -2 & 2 & 1\\ 1 & 2 & -2 \end{array}\right]\ \ \mathbf{Q}^{-1}=\left[\begin{array}{ccc} 0 & 1 & 0\\ -1 & 0 & 0\\ 0 & 0 & -1 \end{array}\right] \end{align*}\]
- \(\mathbf{M}\) is a rotation matrix. \(\left|\mathbf{M}\right|=1\).
Download this page, M5 Special matrices (PDF 170KB)
What's next... M6 Systems of equations