22. Elementary matrices, 1

22. Elementary matrices, 1

We introduce elementary matrices which have the following property: multiplying A on the left by an elementary matrix has the effect of performing a row operation to A .

Elementary matrices of type I

Definition:

Let i,j be integers between 1 and n inclusive, and assume ij . Let λ𝐑 . Then Eij(λ) is the matrix which has:

  • 1s on the diagonal,

  • 0s everywhere else except in the ij th position, where there is a λ .

Such a matrix is an elementary matrix of type I.

For example, if n=3 you get things like: E12(4)=(140010001),E32(-5)=(1000100-51).

If n=4 , you get things like E23(7)=(1000017000100001).

Lemma:

Let A and Eij(λ) be n -by-n matrices. Then Eij(λ)A is the matrix obtained from A by doing RiRi+λRj .

Let's consider the case i<j (the other case is similar so we omit it). Consider the product (1col icol jrow i1λ11)(A11A1nAi1AinAj1AjnAn1Ann)

The only difference the λ makes is when we multiply the i th row into a column of A (say the k th column). Instead of just picking up 1×Aik , we get 1×Aik+λ×Ajk .

In other words, the result Eij(λ)A is obtained from A by adding λ times row j to row i .

Elementary matrices of type II

Definition:

Let i be an integer between 1 and n inclusive and let λ0 be a real number. Define Ei(λ) to be the matrix which has 1s on the diagonal everywhere except in the ii th position, where it has a λ . Such a matrix is an elementary matrix of type II.

If n=3 , E2(λ)=(1000λ0001),E3(-1)=(10001000-1).

Lemma:

Ei(λ)A is obtained from A by performing the row operation RiλRi .

In the product, (1000λ0001)(A11A1nAi1AinAn1Ann)

most of the entries are multiplied by 1 , except those in the i th row of A , which are rescaled by λ .