Elementary Row Operations (初等行变换)

1. Interchange: Interchange two rows.
2. Scaling: Multiply all entries in a row by a nonzero constant.
3. Replacement: Replace one row by the sum of itself and a multiple of another row.

Example

$$\begin{aligned} \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} & \overset{R_1 \leftrightarrow R_2}{\longrightarrow} \begin{bmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{bmatrix} & \overset{R_2 \times 3}{\longrightarrow} \begin{bmatrix} 4 & 5 & 6 \\ 3 & 6 & 9 \\ 7 & 8 & 9 \end{bmatrix} & \overset{R_3 - 2R_2}{\longrightarrow} \begin{bmatrix} 4 & 5 & 6 \\ 3 & 6 & 9 \\ 1 & -4 & -9 \end{bmatrix} \end{aligned}$$

Two matrices are called row equivalent (行等价) if there is a sequence of elementary row operations that transforms one matrix into the other.

Echelon Form (阶梯形)

Some Definitions

A nonzero row or column in a matrix means a row or column that contains at least one nonzero entry.

非零行或列是指至少包含一个非零元素的行或列。

A leading entry of a row refers to the leftmost nonzero entry in a nonzero row.

行的主元素是指一个非零行中最左边的非零元素。

Row Echelon Form (REF)

A rectangular matrix is in row echelon form (阶梯形) (REF) if it has the following properties:

1. All nonzero rows are above any rows of all zeros.
   • $\begin{bmatrix}0 & 2 & 4 \\ 3 & -1 & 0 \\ 0 & 0 & 0\end{bmatrix}$ satisfies this property.
   • $\begin{bmatrix}0 & 0 & 0 \\ 1 & 2 & 3 \\ 0 & 4 & 5\end{bmatrix}$ does not satisfy this property.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
   • $\begin{bmatrix}\color{cyan}{1} & 2 & 0 & 3 \\ 0 & 0 & \color{cyan}{4} & 5 \\ 0 & 0 & 0 & \color{cyan}{6}\end{bmatrix}$ satisfies this property.
   • $\begin{bmatrix}0 & \color{cyan}{2} & 3 \\ \color{red}{4} & 5 & 6 \\ 0 & 0 & \color{cyan}{7}\end{bmatrix}$ does not satisfy this property.
3. All entries in a column below a leading entry are zeros.
   • In fact, any matrix that satisfies property 2 automatically satisfies this property.

Reduced Row Echelon Form (RREF)

If a matrix in echelon form (satisfies all three mentioned properties) satisfies the following additional conditions, then it is in reduced (row) echelon form (简化阶梯形) (RREF):

4. The leading entry in each nonzero row is $1$.
5. Each leading $1$ is the only nonzero entry in its column.
   • $\begin{bmatrix}\color{cyan}{1} & \color{red}{5} & 3 & 0 \\ 0 & \color{cyan}{1} & 2 & 0 \\ 0 & 0 & 0 & \color{cyan}{1} \\ 0 & 0 & 0 & 0\end{bmatrix}$ does NOT satisfies this property.

Each matrix is row equivalent to one and only one reduced echelon matrix (RREF).

Achieving RREF

We can achieve the RREF of a matrix by applying a sequence of elementary row operations. Take the following matrix as an example:

$$\begin{bmatrix} 1 & -2 & 1\\ 3 & 0 & -8\\ 5 & 2 & -5 \end{bmatrix}$$

We can apply a sequence of elementary row operations to transform it into its RREF.

Firstly, we can eliminate the first entry in the rows below the first row:

$$\begin{bmatrix} 1 & -2 & 1\\ 3 & 0 & -8\\ 5 & 2 & -5 \end{bmatrix} \overset{R_2 - 3R_1}{\underset{R_3 - 5R_1}{\longrightarrow}} \begin{bmatrix} 1 & -2 & 1\\ 0 & 6 & -11\\ 0 & 12 & -10 \end{bmatrix}$$

Next, we can eliminate the second entry in the third row:

$$\begin{bmatrix} 1 & -2 & 1\\ 0 & 6 & -11\\ 0 & 12 & -10 \end{bmatrix} \overset{R_3 - 2R_2}{\longrightarrow} \begin{bmatrix} 1 & -2 & 1\\ 0 & 6 & -11\\ 0 & 0 & 14 \end{bmatrix}$$

Finally, we can scale each row to make the leading entries equal to $1$, and then eliminate the entries above each leading $1$:

$$\begin{aligned} \begin{bmatrix} 1 & -2 & 1\\ 0 & 6 & -11\\ 0 & 0 & 14 \end{bmatrix} & \overset{\frac{1}{6}R_2}{\underset{\frac{1}{14}R_3}{\longrightarrow}} \begin{bmatrix} 1 & -2 & 1\\[3pt] 0 & 1 & -\frac{11}{6}\\[3pt] 0 & 0 & 1 \end{bmatrix} \\ & \overset{R_2 + \frac{11}{6}R_3}{\underset{R_1 - R_3}{\longrightarrow}} \begin{bmatrix} 1 & 2 & 0\\[3pt] 0 & 1 & 0\\[3pt] 0 & 0 & 1 \end{bmatrix} \\ & \overset{R_1 - 2R_2}{\longrightarrow} \begin{bmatrix} 1 & 0 & 0\\[3pt] 0 & 1 & 0\\[3pt] 0 & 0 & 1 \end{bmatrix} \end{aligned}$$

Now it's in RREF.

Pivot (主元)

• A pivot position (主元位置) in a matrix $A$ is a location in $A$ that corresponds to a leading $1$ in the reduced echelon form $A$.
• A pivot (主元) is a nonzero number in a pivot position.
• A pivot column (主元列) is a column of $A$ that contains a pivot position.

The number of pivots in a $m \times n$ matrix is at most $\min\{m, n\}$.

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