The standard row operations

I shall now introduce the standard row operations – certain transformations acting on matrices, changing some of their rows while leaving others unchanged. We have already encountered them in some proofs above (for instance, the “downward row additions” in the proof of Theorem 3.63 were one type of row operations), but now we shall give them the systematic treatment they deserve and see them collaborate on producing in Gaussian elimination.

Definition 3.136. For the rest of Chapter 3, we shall use the word “transforma-tion” in a rather specific meaning: Letn ∈_{N. A}transformation of n-rowed matrices will mean a map

{matrices withnrows} → {matrices withnrows}.

In other words, atransformation of n-rowed matricesis a map that transforms each matrix with nrows into a new matrix withnrows.

We shall use the arrow notation for transformations: IfO is some transforma-tion of n-rowed matrices, and ifC and D are two matrices, then we will use the notation “C −→^O D” when we want to say that the transformation O _transforms CintoD. For example, ifMdenotes the transformation of 2-rowed matrices that multiplies each entry of a matrix by 3, then

a b c a⁰ b⁰ c⁰

M

−→

3a 3b 3c 3a⁰ 3b⁰ 3c⁰

. More generally, if O₁,O₂, . . . ,O_m is a sequence of transformations of n-rowed matrices, and if C0,C1, . . . ,Cm are some matrices, then we will use the notation

“C₀ −→^O1 C₁ −→ · · ·^O2 −→^Om Cm” when we want to say that the transposition O_i transforms C_i−1 into C_i for each i ∈ {1, 2, . . . ,m}. Examples for this will appear below once we have defined some actual transformations.

We will also sometimes draw arrows in two directions. Namely: If O and P are two transformations of n-rowed matrices, and if C and D are two matrices, then we will use the notation “C

−→O

←−

P

D” when we want to say that the transfor-mation O _transformsC into Dwhile the transformation P _transforms Dinto C.

This will often happen when two transformations O and P are inverse to each other (i.e., each of them undoes the other).

Definition 3.137. Letn∈ _{N. Let}uandvbe two distinct elements of{1, 2, . . . ,n}. Let λbe a number.

Consider the transformation which transforms an n×m-matrix C (for some m ∈ N) into the product A^λ_u,vC, where A^λ_u,v is the λ-addition matrix as defined in Definition 3.53. It is the transformation that modifies an n×_{m-matrix C by} adding λrowvC to the u-th row (according to Proposition 3.56). This transfor-mation will be called the row addition A^λ_u,v. (We are using the same symbol A^λ_u,v for the transformation and for the λ-addition matrix, since they are so closely related; nevertheless, they are not one and the same thing. I hope that the reader will be able to keep them apart.)

The row addition A^λ_u,v is called adownward row addition if u> v, and is called an upward row additionif u<v.

Note that two matrices C and D (with n rows each) satisfy C ^A

λu,v

−→ _{D if and} only if they satisfyD = A^λ_u,vC. (This is because the row addition A^λ_u,vtransforms each n×m-matrix Cinto A^λ_u,vC.)

Note that the row addition A^λ_u,v is inverse to the row addition A⁻_u,v^λ (since the matrices A^λ_u,vand A⁻_u,v^λ are inverse). Hence, if two matricesCandD(withnrows

Example 3.138. The row addition A⁵_1,3 modifies a 3×m-matrix C by adding 5 row3Cto the 1-st row ofC. Thus, this row addition transforms any 3×4-matrix

 arrow notation, we can write this as follows:



Consider the transformation which transforms an n×m-matrix C (for some m ∈ N) into the product S^λ_uC, where S^λ_u is the λ-scaling matrix as defined in Definition 3.85. It is the transformation that modifies ann×m-matrixC by scal-ing the u-th row by λ (according to Proposition 3.88). This transformation will be called the row scaling S_u^λ. (We are using the same symbol S^λ_u for the transfor-mation and for the λ-scaling matrix. Again, this should not lead to confusion.)

Note that two matrices C and D (with n rows each) satisfy C ^S

λu

−→ D if and only if they satisfy D=S_u^λC. (This is because the row scalingS^λ_u transforms each

n×m-matrix Cinto S^λ_uC.)

Note that the row scalingS_u^λis inverse to the row scalingS^1/λu (since the matri-ces S^λ_u and S^1/λu are inverse). Hence, if two matrices Cand D (with nrows each)

Example 3.140. The row scalingS₂⁵modifies a 3×m-matrixCby scaling the 2-nd row by 5. Thus, this row scaling transforms any 3×2-matrix

 Consider the transformation which transforms an n×m-matrix C (for some m ∈ N) into the product Tu,vC, where Tu,v is the swapping matrix as defined in Definition 3.102. It is the transformation that modifies an n×m-matrix C by swapping the u-th row with the v-th row (according to Proposition 3.105). This transformation will be called the row swap Tu,v. (We are using the same symbol Tu,v for the transformation and for the swapping matrix.)

Note that two matrices C and D (with n rows each) satisfy C −→^Tu,v D if and only if they satisfy D = Tu,vC. (This is because the row swap Tu,v transforms each n×m-matrix Cinto Tu,vC.)

Note that the row swapTu,vis inverse to itself (since the matrices Tu,vand Tu,v

are inverse). Hence, if two matricesCand D(with nrows each) satisfyC −→^Tu,v D, then they satisfyC

Example 3.142. The row swap T_1,3 modifies a 4×m-matrix C by swapping the 1-st row of Cwith the 3-rd row of C. Thus, this row swap transforms any 4×

. Using the arrow notation, we can write this as follows:

For example,







 1 2 3 4 5 6 7 8









T_1,3

−→







 5 6 3 4 1 2 7 8







 .

Definition 3.143. Let n ∈ _{N. The} standard row operations (on matrices with n rows) are:

• row additions (i.e., transformations of the form A^λ_u,v);

• row scalings (i.e., transformations of the form S^λ_u);

• row swaps (i.e., transformations of the form Tu,v).

These row operations allow us to “reduce” any matrix to a certain simple form – called a row echelon form – that has many zeroes (in a sense, it is close to upper-triangular, although the two notions are not exactly the same) and that allows for easily solving linear systems.