Elementary Matrices

Def

An n*n matrix E is called [an elementary matrix] if it can be obtained from the identity matrix $In$ by a single elementary row operation (called the operation corresponding to E).

Operation types

There are three types of operations:

type	operation	inverse
1	interchange 2 rows (p & q)	Interchange q & p
2	multiply a single row by some number k which is non-zero	divide a single row by some number k which is non-zero
3	add k * row to different row.	subtract k * row to a different row

We say that E is of type I, II or III if the operation is of that type.

Ex

$$\begin{array}{r}{E}_1=\left[\begin{array}{cc}0& 1\\ & \\ 1& 0\end{array}\right]\end{array}$$

is elementary of type 1. (b/c it turns into the identity matrix w/ row operation 1)

$$\begin{array}{r}{E}_2=\left[\begin{array}{cc}1& 0\\ & \\ 0& 9\end{array}\right]\end{array}$$

is elementary of type 2, b/c we multiply the identity matrix by 9.

$$\begin{array}{r}{E}_3=\left[\begin{array}{cc}1& 3\\ & \\ 0& 1\end{array}\right]\end{array}$$

is elementary of type 3 b/c we add 3*r2 to row 1.

Suppose now that the matrix

$$\begin{array}{r}A=\left[\begin{array}{ccc}a& b& c\\ & & \\ p& q& r\\ & & \end{array}\right]\end{array}$$

is left multiplied by the above elementary matrices as $E_1,E_2\text{ and }{E}_3$, the retuls are

$$\begin{array}{rl}{E}_{1}A& =\left[\begin{array}{cc}0& 1\\ & \\ 1& 0\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ & & \\ p& q& r\\ & & \end{array}\right]\\ & \\ & =\left[\begin{array}{ccc}p& q& r\\ & & \\ a& b& c\\ & & \end{array}\right]\end{array}$$ $$\begin{array}{rl}{E}_{2}A& =\left[\begin{array}{cc}1& 0\\ & \\ 0& 9\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ & & \\ p& q& r\\ & & \end{array}\right]\\ & \\ & =\left[\begin{array}{ccc}a& b& c\\ & & \\ 9p& 9q& 9r\\ & & \end{array}\right]\end{array}$$ $$\begin{array}{rl}{E}_{3}A& =\left[\begin{array}{cc}1& 3\\ & \\ 0& 1\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ & & \\ p& q& r\\ & & \end{array}\right]\\ & \\ & =\left[\begin{array}{ccc}a+3p& b+3q& c+3r\\ & & \\ p& q& r\end{array}\right]\end{array}$$

So the elementary matrix encodes the operation (type 1,2,3) to be performed.

We observe that, in each case, left multiplying A by the elementary matrix has the same effect as doing the correspdonding operation on A.

Lemma 2.5.1

If an elementary row operation is performaned on an m*n matrix A, the result is just EA where E is the elementary matrix obtained by performing the same operation on [the m*n identity matrix].

Note

The effect of an elementary row operation can be reversed by another such operation (called it’s inverse), which is also elementary of the same type. It follows that each elementary matrix is invertable. In fact, if a row operation on I produces E, then the inverse operation carries E back to I.

Hence, if F is the elementary matrix corresponding to the inverse operation, then $FE=I$ which is to say $F=E^{-1}$

Lemma 2.5.2

Every elementary matrix E is [invertible] and [E-1|notation of previous one] is also an elementary matrix of the same type.

Examples

$$\begin{array}{rl}{E}_1& =\left[\begin{array}{ccc}0& 1& 0\\ & & \\ 1& 0& 0\\ & & \\ 0& 0& 1\\ & & \end{array}\right]\\ & \\ E_2& =\left[\begin{array}{ccc}1& 0& 0\\ & & \\ 0& 1& 0\\ & & \\ 0& 0& 9\end{array}\right]\\ & \\ E_3& =\left[\begin{array}{ccc}1& 0& 5\\ & & \\ 0& 1& 0\\ & & \\ 0& 0& 1\end{array}\right]\end{array}$$

E1 is itself, since it’s the identity matrix. e2 is the same, except the 9 is 1/9 E3 is (same as subract 5*r3 from r1)

\begin{bmatrix} 1 & 0 & -5\\ 0 & 1 & 0\\ 0 & 0 & 1

\end{bmatrix}

Inverses and elementary matrices

Suppose that an m*n matrix is carried to a matrix B (written $A\to B$) by a series of k elementary row operations. Let $E_1,E_2,\dots ,E_k$ denote the corresponding elementary matrices.

By Lemma 2.5.1 the reduction becomes $A\to E_{1}A\to E_2{E}_{1}A\to E_3{E}_2{E}_{1}A\to \cdots \to E_k{E}_{k-1}\dots E_3{E}_2{E}_1=B$ .

In other words, $A\to UA=B$ where $U=E_k{E}_{k-1}\dots E_3{E}_2{E}_1$.

The matrix U is invertible as a product of invertible matrices.

Next, replace A by the identimy matrix $I_m$, we get $I_m\to U{I}_m=U$. Hence, this series of operations carries the block matrix $[A{I}_m]\to [BU]$

Block matrix is the thing we did w/ inversion method.

Theorem 2.5.1

Suppose A is M8N and $A\to B$ by elementary row operations.

1. $B=UA$ where U is an m*m invertible matrix.

2. U can be computed by carrying $[A{I}_m]\to [BU]$

3. $U=E_k{E}_{k-1}\dots E_3{E}_2{E}_1$ where those E’s are elementary matrices.

   j

Ex

if

$$\begin{array}{r}A=\left[\begin{array}{ccc}2& 3& 1\\ & & \\ 1& 2& 1\end{array}\right]\end{array}$$

express the reduced row-echelon form R of A as R=UA where U is an invertible matrix.

$$\left[\begin{array}{cccccc}2& 3& 1& |& 1& 0\\ & & & & & \\ 1& 2& 1& |& 0& 1\\ & & & & & \end{array}\right]$$

r1<=> r2

$$\left[\begin{array}{cccccc}1& 2& 1& |& 0& 1\\ & & & & & \\ 2& 3& 1& |& 1& 0\\ & & & & & \end{array}\right]$$

r2 - r1*2

$$\left[\begin{array}{cccccc}1& 2& 1& |& 0& 1\\ & & & & & \\ 0& -1& -1& |& 1& -2\\ & & & & & \end{array}\right]$$

r2 *= -1

$$\left[\begin{array}{cccccc}1& 2& 1& |& 0& 1\\ & & & & & \\ 0& 1& 1& |& -1& 2\\ & & & & & \end{array}\right]$$

r1 - 2*r2

$$\left[\begin{array}{cccccc}1& 0& -1& |& 2& -3\\ & & & & & \\ 0& 1& 1& |& -1& 2\\ & & & & & \end{array}\right]$$ $$\begin{array}{r}U=\left[\begin{array}{cc}2& -3\\ & \\ -1& 2\end{array}\right]\\ B=\left[\begin{array}{ccc}1& 0& -1\\ & & \\ 0& 1& 1\end{array}\right]\end{array}$$

Note

Suppose A is invertible. We know that $A\to I$ by Theorem 2.4.5. So, taking $B=I$ in theorem 2.5.1, we get $[AI]\to [IU]$ where $I=U\cdot A$, thus $U=A^{-1}$.

So we then have $[AI]\to [I{A}^{-1}]$. This is the same thing as the inversion method.

Now, by theorem 2.5.1, $A^{-1}=U=$ that giant pile of E’s.

where the giant pile of E’ are elementary matrices.

Hence $A=(A^{-1}{)}^{-1}=U^{-1}=(E_k...E_1{)}^{-1}=E_1^{-1}...E_k^{-1}$.

Theorem 2.5.2:

A square matrix is invertible iff it is a product of elementary matrices.

Example 2.5.3

Express

$$\begin{array}{r}A=\left[\begin{array}{cc}-2& 3\\ & \\ 1& 0\end{array}\right]\end{array}$$

Express ^^ as a product of elementary matrices.

$$\left[\begin{array}{ccccc}-2& 3& |& 1& 0\\ & & & & \\ 1& 0& |& 0& 1\\ & & & & \end{array}\right]$$

—

r1<-> r2

$$\left[\begin{array}{ccccc}1& 0& |& 0& 1\\ & & & & \\ -2& 3& |& 1& 0\\ & & & & \end{array}\right]$$

r2 + r1*2

$$\left[\begin{array}{ccccc}1& 0& |& 0& 1\\ & & & & \\ 0& 3& |& 1& 2\\ & & & & \end{array}\right]$$

r2 /= 3

$$\left[\begin{array}{ccccc}1& 0& |& 0& 1\\ & & & & \\ 0& 1& |& 1/3& 2/3\\ & & & & \end{array}\right]$$

In order to generate the relevant E_1’s, you take the identity matrix and apply the row operation to it.

$$\begin{array}{r}{E}_1=\left[\begin{array}{cc}0& 1\\ & \\ 1& 0\end{array}\right]\\ \\ E_2=\left[\begin{array}{cc}1& 0\\ & \\ 2& 1\end{array}\right]\\ \\ E_3=\left[\begin{array}{cc}1& 0\\ & \\ 0& \frac{1}{3}\end{array}\right]\\ \\ U=\left[\begin{array}{cc}0& 1\\ & \\ \frac{1}{3}& \frac{2}{3}\end{array}\right]\end{array}$$