# How to find elementary matrices

2.8: Elementary Matrices

The matrix E = [1 0 0 0 3 0 0 0 1] is the elementary matrix obtained from multiplying the second row of the 3 ? 3 identity matrix by 3. The matrix E = [ 1 0 ? 3 1] is the elementary matrix obtained from adding ? 3 times the first row to the third row. Elementary Matrices and a Method for Finding the Inverse De?nition 1 A n ? n matrix is called an elementary matrix if it can be obtained from In by performing a single elementary row operation Reminder: Elementary row operations: 1. Multiply a row a by k 2 R 2. Exchange two rows 3. Add a multiple of one row to another Theorem 1File Size: 80KB.

We now turn our attention to a special type of matrix called an elementary matrix. An elementary matrix is always a square matrix. Recall the row operations given matrides Definition [def:rowoperations]. You may construct an elementary matrix from any row operation, but remember that you can only apply one operation. Those which involve switching rows of what is a cst test identity matrix are called permutation matrices.

Elementary matrices can be used in place of row operations and therefore are very useful. The following theorem is an important result which we will use throughout this text.

The result is given by. Consider the following lemma. The resulting matrix is given by. There is one last row operation to consider. The following lemma discusses the final operation of adding a multiple of eoementary row to another row. Every elementary matrix is invertible and its inverse is also an elementary matrix. By tracking each row operation completed, this row reduction can be completed through multiplication by elementary matrices. For each step, we will record the appropriate elementary matrix.

This leads to an important consequence related to the above discussion. It follows that.

Introducing the left inverse of a square matrix

The elementarymatrix corresponding to the operation is shown in the right-most column. Matrix. Elementary row operation. Elementary matrix. $$\left[\begin{array}{ccc|c} 1 & 0 & 2 & -1 \\ -2 & 0 & -3 & 1 \\ 0 & 2 & 0 & -2 \end{array}\right]$$ $$R_2 \leftarrow R_2 + 2 R_1$$. Such a matrix is called an elementary matrix. More precisely, we have the following de–nition: De–nition 95 An elementary matrix is an n n matrix which can be ob-tained from the identity matrix I n by performing on I n a single elementary row transformation. Example 96 2 4 0 1 0 1 0 0 0 0 1 3 5 is an elementary matrix. It can be obtained byFile Size: KB.

An elementary matrix is a square matrix that has been obtained by performing an elementary row or column operation on an identity matrix. Remember that there are three types of elementary row operations :. Elementary column operations are defined similarly interchange, addition and multiplication are performed on columns. When elementary operations are carried out on identity matrices they give rise to so-called elementary matrices. Definition A matrix is said to be an elementary matrix if and only if it is obtained by performing an elementary row or column operation on the identity matrix.

Example If we take the identity matrix and multiply its first row by , we obtain the elementary matrix. Example If we take the identity matrix and add twice its second column to the third, we obtain the elementary matrix. Example If we take the identity matrix and interchange its two columns, we obtain the elementary matrix. As we have already explained , elementary matrices can be used to perform elementary operations on other matrices. The following two procedures are equivalent:.

It is possible to represent elementary matrices as rank one updates to the identity matrix. Proposition Any elementary matrix can be written as where and are two column vectors and. Denote by the columns of the identity matrix i. We prove this proposition by showing how to set and in order to obtain all the possible elementary operations.

Let us start from row and column interchanges. Set Then, is a matrix whose entries are all zero, except for the following entries: As a consequence, is the result of interchanging the -th and -th row of the identity matrix or the -th and -th column. Note that , so that. Let us now find how to multiply a row or a column by a non-zero constant. Set Then, is a matrix whose entries are all zero, except for one entry: As a consequence, is the result of multiplying the -th row or column by.

Note that which is different from zero because was assumed to be. Let us now find how to add a multiple of one row or column to another.

Set Then, is a matrix whose entries are all zero, except for one entry: Thus, is the result of adding times the -th row to the -th or adding times the -th column to the -th. In this case, , so that. As we have proved in the lecture on Matrix inversion lemmas , when the condition is satisfied, rank one updates to the identity matrix are invertible and.

Furthermore, the inverse of an elementary matrix is also an elementary matrix. As far as row operations are concerned, this can be seen as follows:. Similar statements are valid for column operations we just need to replace rows with columns in the three points above.

Taboga, Marco Most of the learning materials found on this website are now available in a traditional textbook format. Table of contents Definition Examples How elementary matrices act on other matrices Representation as rank one update Invertibility. The book. Learn more.