Matrix Mathematics
An Introduction for Computer Graphics
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Prof. David Bernstein
James Madison University
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Computer Science Department
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bernstdh@jmu.edu
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Getting Started
- Definition:
- An \(m \times n\) real-valued matrix
consists of \(m\) rows and \(n\)
columns of real numbers
- Some Notation:
- An \(m \times n\) matrix is usually denoted
as follows:
\(\bs{A} =
\left[ \begin{array}{c c c c}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{array} \right]\)
- The set of real-valued \(m \times n\)
matrices is often denoted by
\(\mathbb{R}^{m \times n}\)
- Some Terminology:
- \(a_{ij}\) is called element \(ij\)
- A matrix is said to be square if \(m=n\)
and rectangular otherwise
- Relationship to Vectors:
- A \(1 \times n\) matrix can be thought of as
a row vector and an \(m \times 1\) matrix
can be thought of as a column vector
Some History
- First Uses:
- Introduced by Sylvester in 1850 and Cayley in 1855
to simplify the notation used in the study
of sets of linear equations (i.e., the study
of linear algebra)
- The Notational Advantage:
- Consider the system of linear equations:
- \(y_{1} = a_{11} x_{1} + a_{12} x_{2} + \cdots + a_{1n} x_{n}\)
- \(y_{2} = a_{21} x_{1} + a_{22} x_{2} + \cdots + a_{2n} x_{n}\)
- \(\vdots\)
- \(y_{m} = a_{m1} x_{1} + a_{m2} x_{2} + \cdots + a_{mn} x_{n}\)
- This system can be written as:
- \(
\left[ \begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{m}\end{array}\right] =
\left[ \begin{array}{c c c c}a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}\end{array}\right]
\left[ \begin{array}{c}x_{1} \\ x_{2} \\ \vdots \\ x_{n}\end{array}\right]
\)
- Or, more succinctly, as:
- \(\bs{y} = \bs{A} \bs{x}\)
- Modern Uses:
- Matrices are used in various branches of mathematics
as well as the theoretical and applied sciences
- They are central to the study of computer graphics
Transposition
- Definition:
- The transpose of the matrix \(\bs{A}\), usually
denoted by \(\bs{A}^{\mbox{T}}\), is obtained by
replacing element \(a_{ij}\) with
\(a_{ji}\)
- An Example:
- If
\(\bs{A} =
\left[ \begin{array}{c c c}
5 & 1 & 3\\
8 & 9 & 7\\
6 & 6 & 2\\
5 & 8 & 4
\end{array}\right]
\)
then
\(\bs{A}^{\mbox{T}} =
\left[ \begin{array}{c c c c}
5 & 8 & 6 & 5\\
1 & 9 & 6 & 8\\
3 & 7 & 2 & 4
\end{array}\right]
\)
- Vectors:
- The transpose of a column vector is a row vector
(and vice versa)
Multiplication by a Scalar
- Definition:
- Given \(\lambda \in \mathbb{R}\) and
\(\bs{A} \in \mathbb{R}^{m \times n}\):
-
\(\lambda \bs{A} =
\left[ \begin{array}{c c c c}
\lambda a_{11} & \lambda a_{12} & \cdots & \lambda a_{1n}\\
\lambda a_{21} & \lambda a_{22} & \cdots & \lambda a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
\lambda a_{m1} & \lambda a_{m2} & \cdots & \lambda a_{mn}\end{array}\right]\)
- An Example:
- If
\(\bs{A} =
\left[ \begin{array}{c c c}
5 & 1 & 3\\
8 & 9 & 7\\
6 & 6 & 2\\
5 & 8 & 4\end{array}\right]
\)
then
\(3 \bs{A} =
\left[ \begin{array}{r r r}
15 & 3 & 9\\
24 & 27 & 21\\
18 & 18 & 6\\
15 & 24 & 12\end{array}\right]
\)
Multiplication by a Scalar (cont.)
- Properties:
- \(\lambda (\bs{B} + \bs{C}) = \lambda \bs{B} + \lambda \bs{C}\)
- \((\lambda + \mu)\bs{C} = \lambda \bs{C} + \mu \bs{C}\)
- \((\lambda \mu) \bs{C} = \lambda (\mu \bs{C})\)
- A Warning:
Addition
- Definition:
- Given \(\bs{A} \in \mathbb{R}^{m \times n}\) and
\(\bs{B} \in \mathbb{R}^{m \times n}\):
-
\(\bs{A} + \bs{B} =
\left[ \begin{array}{c c c c}
a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+a_{1n}\\
a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{mn}+a_{mn}\end{array}\right] \)
- An Example:
- If
\(\bs{A} =
\left[ \begin{array}{r r r}
5 & 1 & 3\\
8 & 9 & 7\\
6 & 6 & 2\\
5 & 8 & 4\end{array}\right]
\)
and
\(\bs{B} =
\left[ \begin{array}{r r r}
2 & 7 & 1\\
-1 & 4 & 6\\
3 & -6 & 7\\
1 & 1 & 1\end{array}\right]
\)
then
\(\bs{A} + \bs{B} =
\left[ \begin{array}{r r r}
7 & 8 & 4\\
7 & 13 & 13\\
9 & 0 & 9\\
6 & 9 & 5\end{array}\right]
\)
- Be Careful:
- You can only add two matrices if they are the same size
(i.e., have the same dimensionality)
Subtraction
- Definition:
- Given \(\bs{A} \in \mathbb{R}^{m \times n}\) and
\(\bs{B} \in \mathbb{R}^{m \times n}\):
-
\(\bs{A} - \bs{B} =
\left[ \begin{array}{c c c c}
a_{11}-b_{11} & a_{12}-b_{12} & \cdots & a_{1n}-a_{1n}\\
a_{21}-b_{21} & a_{22}-b_{22} & \cdots & a_{2n}-a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1}-b_{m1} & a_{m2}-b_{m2} & \cdots & a_{mn}-a_{mn}\end{array}\right] \)
- An Example:
- If
\(\bs{A} =
\left[ \begin{array}{r r r}
5 & 1 & 3\\
8 & 9 & 7\\
6 & 6 & 2\\
5 & 8 & 4\end{array}\right]
\)
and
\(\bs{B} =
\left[ \begin{array}{r r r}
2 & 7 & 1\\
-1 & 4 & 6\\
3 & -6 & 7\\
1 & 1 & 1\end{array}\right]
\)
then
\(\bs{A} - \bs{B} =
\left[ \begin{array}{r r r}
3 & -6 & 2\\
9 & 5 & 1\\
3 & 12 & -5\\
4 & 7 & 3\end{array}\right]
\)
- Be Careful:
- You can only add two matrices if they are the same size
(i.e., have the same dimensionality)
Multiplication by a Vector (cont.)
- Properties:
- \(\bs{A}(\lambda \bs{v}) = \lambda A\bs{v}\)
- \(\bs{A}(\bs{u}+\bs{v}) = \bs{A}\bs{u} + \bs{A}\bs{v}\) (Distributive Law)
- A (Repeated) Warning:
Multiplication (cont.)
- An Observation:
- In general, \(\bs{A} \bs{B}\) does not equal
\(\bs{B} \bs{A}\) (i.e., matrix multiplication
is not commutative)
- An Example:
- If
\(\bs{A} =
\left[ \begin{array}{r r}
1 & 2\\
3 & 4\end{array}\right]
\)
and
\(\bs{B} =
\left[ \begin{array}{r r}
0 & -1\\
6 & 7\end{array}\right]
\)
then
\(\bs{A} \bs{B} =
\left[ \begin{array}{r r}
1 \cdot 0 + 2 \cdot 6 & 1 \cdot -1 + 2 \cdot 7\\
3 \cdot 0 + 4 \cdot 6 & 3 \cdot -1 + 4 \cdot 7\end{array}\right] =
\left[ \begin{array}{r r}
12 & 13\\
24 & 25\end{array}\right]
\)
and
\(\bs{B} \bs{A} =
\left[ \begin{array}{r r}
0 \cdot 1 - 1 \cdot 3 & 0 \cdot 2 - 1 \cdot 4\\
6 \cdot 1 + 7 \cdot 3 & 6 \cdot 2 + 7 \cdot 4\end{array}\right] =
\left[ \begin{array}{r r}
-3 & -4\\
27 & 40\end{array}\right]
\)
The Null Matrix
- Definition:
- A null matrix (or zero matrix)
is an \(m \times n\)
matrix in which each element is 0
- Notation:
- Null matrices are commonly denoted by \(\bs{0}\)
- An Example:
-
\(\bs{0} =
\left[ \begin{array}{c c c c}
0 & 0 & \cdots & 0\\
0 & 0 & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & 0\end{array}\right]\)
- Two Observations:
- Given any matrix \(\bs{A} \in \mathbb{R}^{m \times n}\)
and the null matrix \(\bs{0} \in \mathbb{R}^{m \times n}\)
it follows that
\(\bs{A} + \bs{0} = \bs{0} + \bs{A} = \bs{A}\)
- Given any matrix \(\bs{A} \in \mathbb{R}^{m \times n}\)
and the appropriately sized null matrices,
it follows that
\(\begin{array}{c}
\mbox{ }\\ \mbox{ }\bs{A}\mbox{ }\\ (m \times n)
\end{array}
\begin{array}{c}
\mbox{ }\\ \mbox{ }\bs{0}\mbox{ }\\ (n \times p)
\end{array}
=
\begin{array}{c}
\mbox{ }\\ \mbox{ }\bs{0}\mbox{ }\\ (m \times p)
\end{array}\)
and
\(\begin{array}{c}\mbox{ } \\ \mbox{ }\bs{0}\mbox{ } \\ (p \times m) \end{array}
\begin{array}{c}\mbox{ } \\ \mbox{ }\bs{A}\mbox{ } \\ (m \times n) \end{array} =
\begin{array}{c}\mbox{ } \\ \mbox{ }\bs{0}\mbox{ } \\ (p \times n) \end{array}\)
Division
- Of Scalars:
- Given \(a \in \mathbb{R}\) and \(b \in \mathbb{R}_{++}\)
the quotient \(a/b\) can be written as
either \(a b^{-1}\) or \(b^{-1} a\)
(where \(b^{-1}\) denotes the reciprocal or inverse)
- Of Matrices:
- Suppose, for the moment, that given
\(\bs{B} \in \mathbb{R}^{n \times n}\)
there is a notion of an inverse
\(\bs{B}^{-1} \in \mathbb{R}^{n \times n}\)
and that the inverse exists
- It may not be the case that \(\bs{A} \bs{B}^{-1}\) equals
\(\bs{B}^{-1} \bs{A}\)
- Hence, division cannot be defined without ambiguity
The Identity Matrix
- Definition:
- An identity matrix is an \(n \times n\)
matrix in which the \((i,j)\)th element is 1
when \(i = j\) and 0 otherwise
- Notation:
- Identity matrices are commonly denoted by either
\(\bs{I}\) or \(\bs{1}\)
- An Example:
-
\(\bs{I} =
\left[ \begin{array}{c c c c}
1 & 0 & \cdots & 0\\
0 & 1 & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & 1\end{array}\right]\)
- An Observation:
- Given any matrix \(\bs{A} \in \mathbb{R}^{n \times n}\)
and the identity matrix
\(\bs{I} \in \mathbb{R}^{n \times n}\)
it follows that \(\bs{A} \bs{I} = \bs{I} \bs{A} = \bs{A}\)
Inverse
- Definition:
- The matrix \(\bs{A}\) is said to be invertible if there
exists a matrix \(\bs{A}^{-1}\), called its
inverse, such that
\(\bs{A} \bs{A}^{-1} = \bs{A}^{-1} \bs{A} = \bs{I}\).
- Notes:
- Not all matrices are invertible
- Inverses are fairly difficult to calculate
Addition and Multiplication
- Properties:
- \(\bs{A}+\bs{B} = \bs{B}+\bs{A}\) (Commutative Law of Addition)
- \(\bs{A}+(\bs{B}+\bs{C}) = (\bs{A}+\bs{B})+\bs{C}\) (Associative Law of Addition)
- \(\bs{A}(\bs{B}\bs{C}) = (\bs{A}\bs{B})\bs{C}\) (Associative Law of Multiplication)
- \(\bs{A}(\bs{B}+\bs{C}) = \bs{A}\bs{B} + \bs{A}\bs{C}\) (Distributive Law)
- \((\bs{B}+\bs{C})\bs{A} = \bs{B}\bs{A} + \bs{C}\bs{A}\) (Distributive Law)
- \((\bs{A} + \bs{B})^{\mbox{T}} = \bs{A}^{\mbox{T}} + \bs{B}^{\mbox{T}}\)
- \((\bs{A}\bs{B})^{\mbox{T}} = \bs{B}^{\mbox{T}} \bs{A}^{\mbox{T}}\)
- A Warning (One More Time):
Addition and Multiplication (cont.)
- One Surprising Result:
- \(\bs{A} \bs{B} = \bs{0}\) does not imply \(\bs{A}=\bs{0}\)
or \(\bs{B}=\bs{0}\)
- \(\left[ \begin{array}{r r}2 & 4\\1 & 2\end{array}\right] \left[ \begin{array}{r r}-2 & 4\\1 & -2\end{array}\right] = \left[ \begin{array}{r r}0 & 0\\0 & 0\end{array}\right]
\)
- Another Surprising Result:
- \(\bs{A} \bs{B} = \bs{A} \bs{C}\) does not imply
\(\bs{B}=\bs{C}\)
- \(\left[ \begin{array}{r r}2 & 3\\6 & 9\end{array}\right] \left[ \begin{array}{r r} 1 & 1\\1 & 2\end{array}\right] =
\left[ \begin{array}{r r}2 & 3\\6 & 9\end{array}\right] \left[ \begin{array}{r r}-2 & 1\\3 & 2\end{array}\right]
\)
Determinants of 2x2 Matrices (cont.)
- Understanding the Visualization:
- Areas of the Colored Triangles:
- Blue: \(\frac{1}{2} q_{1} q_{2}\)
- Red: \(\frac{1}{2} r_{1} r_{2}\)
- Green: \(\frac{1}{2} (q_{1}-r_{1})(r_{2} - q_{2}) =
\frac{1}{2} (q_{1} r_{2} - q_{1} q_{2} -r_{1} r_{2} + r_{1} q_{2})
\)
- Area of the White Triangle:
- \(
\begin{align}
& q_{1} r_{2} - \frac{1}{2} q_{1} q_{2} - \frac{1}{2} r_{1} r_{2} -
\frac{1}{2} q_{1} r_{2} + \frac{1}{2} q_{1} q_{2} + \frac{1}{2} r_{1} r_{2} - \frac{1}{2} r_{1} q_{2} \\
= & \frac{1}{2} q_{1} r_{2} - \frac{1}{2} r_{1} q_{2} \\
= & \frac{1}{2} (q_{1} r_{2} - r_{1} q_{2}) = \frac{1}{2} |\bs{A}|
\end{align}
\)
Determinants of 2x2 Matrices (cont.)
- A Numerical Example:
- \(\bs{A} = \left[ \begin{array}{r r}q_{1} & r_{1}\\ q_{2} & r_{2}\end{array}\right] = \left[ \begin{array}{r r}1.5 & 0.5\\ 0.5 & 1.0\end{array}\right]\)
- \(\frac{1}{2} |\bs{A}| =
\frac{1}{2} (1.5 \cdot 1.0 - 0.5 \cdot 0.5) =
\frac{1}{2} 1.25 = 0.625\)
- A Related Example:
- \(\bs{B} = \left[ \begin{array}{r r}r_{1} & q_{1}\\ r_{2} & q_{2}\end{array}\right] = \left[ \begin{array}{r r}0.5 & 1.5\\ 1.0 & 0.5\end{array}\right]\)
- \(\frac{1}{2} |\bs{B}| =
\frac{1}{2} (0.5 \cdot 0.5 - 1.0 \cdot 1.5) =
\frac{1}{2} -1.25 = -0.625\)
Determinants of 2x2 Matrices (cont.)
- An Observation:
- The determinant can be positive or negative (or 0)
- Signed Area (The Right Hand Rule):
- If, using your right hand, you can curl your fingers from
the first column/point to the second column/point then
the area should be positive.
- This is often sometimes called the counter-clockwise rule
Determinants of Other Square Matrices (cont.)
- A \(3 \times 3\) Matrix:
-
\(\bs{A} = \begin{bmatrix}
6 & 1 & 1 \\
4 & -2 & 5 \\
2 & 8 & 7 \\
\end{bmatrix}\)
- The Determinant:
-
\(
|\bs{A}| =
+ 6 \cdot \left| \begin{bmatrix}-2 & -5 \\ 8 & 7 \end{bmatrix} \right|
- 1 \cdot \left| \begin{bmatrix} 4 & 5 \\ 2 & 7 \end{bmatrix} \right|
+ 1 \cdot \left| \begin{bmatrix} 4 & -2 \\ 2 & 8 \end{bmatrix} \right|
\\ = 6 \cdot (-2 \cdot 7 - 5 \cdot 8)
- 1 \cdot (4 \cdot 7 - 5 \cdot 2)
+ 1 \cdot (4 \cdot 8 - (-2) \cdot 2) \\
= 6 \cdot (-54) - 1 \cdot (18) + 1 \cdot (36) = -306
\)
Determinants
- Properties:
- \(|\lambda \bs{A}| = \lambda^{n} |\bs{A}|\)
- \(|\bs{A} \bs{B}| = |\bs{A}| \cdot |\bs{B}|\)
- \(|\bs{A}| = |\bs{A}^{\mbox{T}}|\)
- An Interesting Example:
There's Always More to Learn
