Elementary Transformations and the Rank of a Matrix

Elementary Transformations

Elementary Row Operations

The following three transformations of a matrix are called elementary row operations:

1. Interchanging two rows: interchange row $i$ and row $j$, written $R_i \leftrightarrow R_j$
2. Multiplying a row by a scalar: multiply row $i$ by a nonzero constant $k$, written $R_i \times k$
3. Adding a multiple of one row to another: add $k$ times row $j$ to row $i$, written $R_i + kR_j$

Elementary Column Operations

The following three transformations of a matrix are called elementary column operations:

1. Interchanging two columns: interchange column $i$ and column $j$, written $C_i \leftrightarrow C_j$
2. Multiplying a column by a scalar: multiply column $i$ by a nonzero constant $k$, written $C_i \times k$
3. Adding a multiple of one column to another: add $k$ times column $j$ to column $i$, written $C_i + kC_j$

Properties of Elementary Transformations

1. Invertibility: each elementary transformation has its own inverse transformation
2. Rank preservation: elementary transformations do not change the rank of the matrix
3. Equivalence preservation: elementary transformations preserve the equivalence relation between matrices

Elementary Matrices

The Definition of an Elementary Matrix

A matrix obtained by applying one elementary transformation to the identity matrix $I$ is called an elementary matrix.

The Types of Elementary Matrices

1. Interchange Elementary Matrices

The matrix obtained by interchanging row $i$ and row $j$ of the identity matrix:

$$E_{ij} = \begin{pmatrix} 1 & & & & & & \\ & \ddots & & & & & \\ & & 0 & & 1 & & \\ & & & 1 & & & \\ & & 1 & & 0 & & \\ & & & & & \ddots & \\ & & & & & & 1 \end{pmatrix}$$

2. Scaling Elementary Matrices

The matrix obtained by multiplying row $i$ of the identity matrix by a nonzero constant $k$:

$$E_i(k) = \begin{pmatrix} 1 & & & & \\ & \ddots & & & \\ & & k & & \\ & & & \ddots & \\ & & & & 1 \end{pmatrix}$$

3. Shear Elementary Matrices

The matrix obtained by adding $k$ times row $j$ of the identity matrix to row $i$:

$$E_{ij}(k) = \begin{pmatrix} 1 & & & & & \\ & \ddots & & & & \\ & & 1 & & k & \\ & & & \ddots & & \\ & & & & 1 & \\ & & & & & \ddots \end{pmatrix}$$

Properties of Elementary Matrices

1. Invertibility: every elementary matrix is invertible

2. Inverse matrix:

   • $E_{ij}^{-1} = E_{ij}$
   • $E_i(k)^{-1} = E_i(\frac{1}{k})$
   • $E_{ij}(k)^{-1} = E_{ij}(-k)$

3. Left-multiplication: left-multiplying a matrix $A$ by an elementary matrix amounts to performing the corresponding elementary row operation on $A$

4. Right-multiplication: right-multiplying a matrix $A$ by an elementary matrix amounts to performing the corresponding elementary column operation on $A$

The Rank of a Matrix

The Definition of the Rank of a Matrix

The rank of a matrix $A$ is the number of linearly independent rows (or columns) in the largest such set, written $r(A)$ or $\text{rank}(A)$.

Equivalent definitions:

• The rank of $A$ equals the order of its highest-order nonzero minor
• The rank of $A$ equals the number of nonzero rows in its row echelon form

Properties of the Rank of a Matrix

1. Basic properties:

   • $0 \leq r(A) \leq \min(m, n)$ ($A$ is an $m \times n$ matrix)
   • $r(A) = 0$ if and only if $A = O$
   • $r(A) = n$ if and only if $A$ has full rank

2. Properties under operations:

   • $r(A + B) \leq r(A) + r(B)$
   • $r(AB) \leq \min(r(A), r(B))$
   • $r(A^T) = r(A)$
   • $r(kA) = r(A)$ ($k \neq 0$)

3. Elementary transformation property: elementary transformations do not change the rank of the matrix

Methods for Computing the Rank of a Matrix

Method 1: The Minor Method

1. Starting from the highest order, compute the minors of the matrix
2. The first nonzero minor you find has order equal to the rank of the matrix

Method 2: The Elementary Transformation Method

1. Apply elementary row operations to the matrix to reduce it to row echelon form
2. Count the number of nonzero rows

Example: find the rank of the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 1 \end{pmatrix}$

Solution: $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 1 \end{pmatrix}$

$R_2 - 2R_1$: $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{pmatrix}$

$R_3 - R_1$: $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & -1 & -2 \end{pmatrix}$

$R_2 \leftrightarrow R_3$: $\begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & -2 \\ 0 & 0 & 0 \end{pmatrix}$

There are 2 nonzero rows, so $r(A) = 2$.

Matrix Equivalence

The Definition of Matrix Equivalence

If the matrix $A$ can be transformed into the matrix $B$ by a finite number of elementary transformations, then $A$ and $B$ are called equivalent, written $A \sim B$.

The Necessary and Sufficient Condition for Matrix Equivalence

The Canonical Form

Every matrix can be transformed by elementary transformations into the following form:

$$\begin{pmatrix} I_r & O \\ O & O \end{pmatrix}$$

where $I_r$ is the r-th order identity matrix and $r$ is the rank of the matrix.

Block Matrices

The Definition of a Block Matrix

Splitting a matrix along its rows and columns into several submatrices gives these submatrices the name block matrices.

Example:

$$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$$

where $A_{11}, A_{12}, A_{21}, A_{22}$ are all submatrices.

Operations on Block Matrices

1. Addition of Block Matrices

Let $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$ and $B = \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix}$. Then:

$A + B = \begin{pmatrix} A_{11} + B_{11} & A_{12} + B_{12} \\ A_{21} + B_{21} & A_{22} + B_{22} \end{pmatrix}$

2. Multiplication of a Block Matrix by a Scalar

$kA = \begin{pmatrix} kA_{11} & kA_{12} \\ kA_{21} & kA_{22} \end{pmatrix}$

3. Multiplication of Block Matrices

Let $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$ and $B = \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix}$. Then:

$AB = \begin{pmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{pmatrix}$

Note: multiplication of block matrices requires the corresponding submatrices to be multipliable.

Special Block Matrices

1. Block-Diagonal Matrices

$$A = \begin{pmatrix} A_1 & & \\ & A_2 & \\ & & \ddots \\ & & & A_n \end{pmatrix}$$

2. Upper-Triangular Block Matrices

$$A = \begin{pmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\ & A_{22} & \cdots & A_{2n} \\ & & \ddots & \vdots \\ & & & A_{nn} \end{pmatrix}$$

3. Lower-Triangular Block Matrices

$$A = \begin{pmatrix} A_{11} & & & \\ A_{21} & A_{22} & & \\ \vdots & \vdots & \ddots & \\ A_{n1} & A_{n2} & \cdots & A_{nn} \end{pmatrix}$$

The Determinant of a Block Matrix

For a $2 \times 2$ block matrix $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$:

1. If $A_{11}$ is invertible, then: $|A| = |A_{11}| \cdot |A_{22} - A_{21}A_{11}^{-1}A_{12}|$

2. If $A_{22}$ is invertible, then: $|A| = |A_{22}| \cdot |A_{11} - A_{12}A_{22}^{-1}A_{21}|$

3. If $A_{12} = O$ or $A_{21} = O$, then: $|A| = |A_{11}| \cdot |A_{22}|$

Exercises

Exercise 1

Write down the elementary matrix that interchanges the first row and the second row of the matrix $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

Exercise 2

Reduce the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 1 \end{pmatrix}$ to row echelon form by elementary row operations.

Exercise 3

Find the rank of the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 1 \end{pmatrix}$.

Exercise 4

Decide whether the matrices $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}$ are equivalent.

Exercise 5

Compute the determinant of the block matrix $\begin{pmatrix} A & B \\ O & C \end{pmatrix}$, where both $A$ and $C$ are invertible matrices.

Exercise 6

Prove that: if $A$ is an $m \times n$ matrix, then $r(A) \leq \min(m, n)$.

Summary

Symbols Used in This Article