5.4: Matrix Notation
- Page ID
- 63882
Review Sections 6.1 - 6.3 of the Stephen Boyd and Lieven Vandenberghe Applied Linear algebra book which introduces the concept of Matrices. Some things to take away include:
- Basic Matrix composition
- Transpose Addition and norms
- Zero and Identy matrix
A matrix is a rectangular array of numbers typically written between rectangular brackets such as:
\[ \begin{split} A = \left[ \begin{matrix} 0 & -1 \\ 3 & 4 \\ 0 & 2 \end{matrix} \right]^{ 3\times 2} \end{split} \nonumber \]
The 3×2 subscript is not always included but is handy notation to remember the size of a matrix. The size of a matrix is always written \(m \times n\) where \(m\) is the number of rows and \(n\) is the number of columns. So in the above case Matrix \(A\) is a 3×2 (read “three by two”) matrix.
What is the size of the following matrix?
\[ \begin{split} B = \left[ \begin{matrix} 0 & -1 & 0 \\ 3 & 4 & 2 \\ \end{matrix} \right] \end{split} \nonumber \]
Each element in a matrix can be referenced by it’s index location. Similar to the size of a matrix the location of an element is described by two numbers, it’s row followed by it’s column. Counting for the rows start at the top and the columns start on the left. For example, in Matrix \(B\) element \(b_{1,2}\) is the number in row 1 column 2 which is -1.
What is the value of element (2,1) in matrix \(B\)?
A linear system of equations can be written in matrix format. For example, the equations in the original example can be written as the following “Augmented matrix”
\[ \begin{split}
\left[
\begin{matrix}
1 & 1 \\
20 & 25
\end{matrix}
\, \middle\vert \,
\begin{matrix}
30 \\
690
\end{matrix}
\right]
\end{split} \nonumber \]
And the example which included silversmith can be written as follows:
\[ \left[
\begin{matrix}
1 & 1 & 1 \\
50 & 20 & 25 \\
110 & 0 & 20
\end{matrix}
\, \middle\vert \,
\begin{matrix}
30 \\
690 \\
300
\end{matrix}
\right] \nonumber \]
The above equations are represented as “augmented matrices” with the equal side represented as a vertical line.
The general matrix format for a system of linear equations can be written as follows:
\[ \begin{split}
X =
\left[
\begin{matrix}
x_{11} & x_{12} & x_{13} & \dots \\
x_{21} & x_{22} & x_{23} & \dots \\
\ldots & \ldots & \ldots & \ddots \\
x_{m1} & x_{m2} & x_{m3} & \dots
\end{matrix}
\, \middle\vert \,
\begin{matrix}
x_{1n} \\ x_{2n} \\ \ldots \\ x_{mn}
\end{matrix}
\right] ^{mxn}
\end{split} \nonumber \]
where \(x_{ij}\) is a scalar element in the matrx.
Now consider the following system of linear equations:
\[x_1 = 2.14159 \nonumber \]
\[x_2 = 4 \nonumber \]
\[x_3 = -7.2 \nonumber \]
\[x_4 = 69 \nonumber \]
\[x_5 = 84 \nonumber \]
\[x_6 = 240 \nonumber \]
Lets rewrite this equation as an augmented matrix:
\[ \begin{split}
X =
\left[
\begin{matrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{matrix}
\, \middle\vert \,
\begin{matrix}
2.14159 \\ 4 \\ -7.2 \\ 69 \\ 84 \\ 240
\end{matrix}
\right] ^{6x7}
\end{split} \nonumber \]
Notice the submatrix on the left hand side is just the \(I_{6}\) identity matrix and the right hand side are the solutions.