MATH 323 Lecture 6

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Elementary Matrices

System of equations $A\overrightarrow{x}=\overrightarrow{b}$ can be converted to $U\overrightarrow{x}=\overrightarrow{c}$, where $U=\left(\prod _{i=k}^1{E}_i\right)A$ and $\overrightarrow{c}=\left(\prod _{i=k}^1{E}_i\right)\overrightarrow{b}$. $E_i$ is an elementary matrix of type 1, 2, or 3.

Properties:

• Premultiplication performs rule operations on rows
• Postmultiplication performs rule operations on columns
• $E^{-1}$ is an elementary matrix of the same type.

Row Equivalence

describes that a matrix $A$ can be converted to another matrix $B$ by applying elementary matrices: $B=\left(\prod _{i=k}^1{E}_i\right)A$

If $A$ is nonsingular, then $A$ is row equivalent to $I$: $\begin{array}{rl}I& =\left(\prod _{i=k}^1{E}_i\right)A=A{A}^{-1}\\ A^{-1}& =\prod _{i=k}^1{E}_i\end{array}$

Therefore, the augmented matrix $\left[\begin{array}{cc}A& I\end{array}\right]$ can be converted to $\left[\begin{array}{cc}I& A^{-}1\end{array}\right]$ by performing elementary row operations to convert it to reduced row echelon form.

Example:

$\begin{array}{rl}\left[\begin{array}{cccccc}1& 4& 3& 1& 0& 0\\ -1& -2& 0& 0& 1& 0\\ 2& 2& 3& 0& 0& 1\end{array}\right]& \to \left[\begin{array}{cccccc}1& 0& 0& -\frac{1}{2}& -\frac{1}{2}& \frac{1}{2}\\ 0& 1& 0& \frac{1}{4}& -\frac{1}{4}& -\frac{1}{4}\\ 0& 0& 1& \frac{1}{6}& \frac{1}{2}& \frac{1}{6}\end{array}\right]\\ A& =\left[\begin{array}{ccc}1& 4& 3\\ -1& -2& 0\\ 2& 2& 3\end{array}\right]\\ A^{-1}& =\left[\begin{array}{ccc}-\frac{1}{2}& -\frac{1}{2}& \frac{1}{2}\\ \frac{1}{4}& -\frac{1}{4}& -\frac{1}{4}\\ \frac{1}{6}& \frac{1}{2}& \frac{1}{6}\end{array}\right]\end{array}$

Therefore, if we have the system of equations $A\overrightarrow{x}=\overrightarrow{b}$, the solution is $\overrightarrow{x}=A^{-1}\overrightarrow{b}$.

Triangular Matrices

Suppose $A$ is a $n\times n$ square matrix.

• If $A$ is of the form $\left(\begin{array}{cccc}{x}_1& 0& \dots & 0\\ 0& \ddots & 0& ⋮\\ ⋮& 0& \ddots & 0\\ 0& \dots & 0& x_n\end{array}\right)$, it is diagonal.
• If $A$ is of the form $\left(\begin{array}{cccc}{x}_1& {\alpha }_2& \dots & \beta \\ 0& \ddots & 0& ⋮\\ ⋮& 0& \ddots & \gamma \\ \delta & 0& 0& x_n\end{array}\right)$, it is upper triangular.
• If $A$ is of the form $\left(\begin{array}{cccc}{x}_1& 0& \dots & 0\\ \alpha & \ddots & 0& ⋮\\ ⋮& \beta & \ddots & 0\\ \gamma & \dots & \delta & x_n\end{array}\right)$, it is lower triangular.

Triangular Factorization

If a $n\times n$ matrix $A$ can be reduced to strict upper triangular form using row operations of type 3, then $A=\mathrm{\angle }U$, where $\mathrm{\angle }$ – unit lower triangular matrix and $U$ – strictly upper triangular matrix.

Each operation is of the form $(\mathrm{\#})-{\ell }_{ij}(\mathrm{\#})$, where $i$ and $j$ are row indices of $A$.

$A=\left[\begin{array}{ccc}2& 4& 2\\ 1& 5& 2\\ 4& 1& 9\end{array}\right]⟶U=\left[\begin{array}{ccc}2& 4& 2\\ 0& 3& 1\\ 0& 0& 8\end{array}\right]$

${\ell }_{21}=\frac{1}{2}$, ${\ell }_{31}=2$, and ${\ell }_{32}=-3$.

If we construct a $n\times n$ matrix by replacing elements of $I_n$ with corresponding $\ell$ values, we get $L=\left[\begin{array}{ccc}1& 0& 0\\ -\frac{1}{2}& 1& 0\\ 2& -3& 1\end{array}\right]$ such that $A=LU$ and $L=\prod _{i=k}^1{E}_i^{-1}$, where $E_i^{-1}$ is a single $\ell$ replacement in $I$ for each step.

Determinants

For a $n\times n$ square matrix $A$, there is a corresponding scalar determinant of $A$

Written in any of the following formats:

$\det (A)=\left|A\right|=\left|\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{n1}& \dots & a_{nn}\end{array}\right|$

A matrix is singular iff its determinant is 0.

1. For a $1\times 1$ square matrix, $|A|=a$
2. For a $2\times 2$ square matrix, $|A|=a_{11}{a}_{22}-a_{12}{a}_{21}$ (product of main diagonal minus product of "secondary diagonal")
3. For a $3\times 3$ square matrix, $|A|=a_{11}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|-a_{22}\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|+a_{13}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$ (See MATH 251 Lecture 10 for hints and tricks)