Identity Matrix from Upper Triangular Matrix
Theorem
Let $\mathbf A = \sqbrk a_{m n}$ be an upper triangular matrix of order $m \times n$ with no zero diagonal elements.
Let $k = \min \set {m, n}$.
Then $\mathbf A$ can be transformed into a matrix such that the first $k$ rows and columns form the unit matrix of order $k$.
Proof
By definition of $k$:
- if $\mathbf A$ has more rows than columns, $k$ is the number of columns of $\mathbf A$.
- if $\mathbf A$ has more columns than rows, $k$ is the number of rows of $\mathbf A$.
Thus let $\mathbf A'$ be the square matrix of order $k$ consisting of the first $k$ rows and columns of $\mathbf A$:
- $\mathbf A' = \begin {bmatrix}
a_{1 1} & a_{1 2} & a_{1 3} & \cdots & a_{1, k - 1} & a_{1 k} \\
0 & a_{2 2} & a_{2 3} & \cdots & a_{2, k - 1} & a_{2 k} \\ 0 & 0 & a_{3 3} & \cdots & a_{3, k - 1} & a_{3 k} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & a_{k - 1, k - 1} & a_{k - 1, k} \\ 0 & 0 & 0 & \cdots & 0 & a_{k k} \\
\end {bmatrix}$
$\mathbf A$ can be transformed into echelon form $\mathbf B$ by using the elementary row operations:
- $\forall j \in \set {1, 2, \ldots, k}: e_j := r_j \to \dfrac 1 {a_{j j} } r_j$
Again, let $\mathbf B'$ be the square matrix of order $k$ consisting of the first $k$ rows and columns of $\mathbf B$:
- $\mathbf B' = \begin {bmatrix}
1 & b_{1 2} & b_{1 3} & \cdots & b_{1, k - 1} & b_{1 k} \\ 0 & 1 & b_{2 3} & \cdots & b_{2, k - 1} & b_{2 k} \\ 0 & 0 & 1 & \cdots & b_{3, k - 1} & b_{3 k} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 1 & b_{k - 1, k} \\ 0 & 0 & 0 & \cdots & 0 & 1 \\
\end {bmatrix}$
$\mathbf B$ is then transformed into reduced echelon form $\mathbf C$ by means of the elementary row operations:
- $\forall j \in \set {1, 2, \ldots, k - 1}: e_{j k} := r_j \to r_j - b_{j k} r_k$
- $\forall j \in \set {1, 2, \ldots, k - 2}: e_{j, k - 1} := r_j \to r_j - b_{j, k - 1} r_{k - 1}$
and so on, until:
- $e_{1 2} := r_1 \to r_1 - b_{1 2} r_2$
Again, let $\mathbf C'$ be the square matrix of order $k$ consisting of the first $k$ rows and columns of $\mathbf C$:
- $\mathbf C' = \begin {bmatrix}
1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \\
\end {bmatrix}$
By inspection, $\mathbf C$ is seen to be the unit matrix of order $k$.
$\blacksquare$
Examples
Arbitrary Matrix 1
Let $\mathbf A$ be the matrix defined as:
- $\mathbf A = \begin {bmatrix}
1 & 1 & 1 & 1 \\ 0 & 2 & 2 & 2 \\ 0 & 0 & 3 & 4 \\ \end {bmatrix}$
is row equivalent to the reduced echelon matrix:
- $\mathbf E = \begin {bmatrix}
1 & 0 & 0 & -\dfrac 1 2 \\ 0 & 1 & 0 & \dfrac 1 6 \\ 0 & 0 & 1 & \dfrac 4 3 \\ \end {bmatrix}$
Sources
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations