The content in today's blog is taken from Hans Schneider and George Phillip Barker's Matrices and Linear Algebra.
Definition 1: Type I Column Operation A(Ci ↔ Cj)
A Type I column operation consists of the interchange of any two distinct columns.
Definition 2: Type II Column Operation A(kCi) where k ≠ 0.
A Type II column operation consists of the multiplication of any column by a nonzero scalar.
Definition 3: Type III Column Operation A(kCi + Cj).
A Type III column operation consists of a scalar multiple of some column to another column.
Definition 4: Elementary Column Operation
An elementary column operation is any Type I, Type II, or Type III Column Operation or any combination of these operations.
Definition 5: Column Equivalence
Two matrices A,B are column equivalent if and only if A can be obtained from B by elementary column operations.
Lemma 1:
The interchange of two columns in a given matrix A is equivalent to a multiplication between a matrix A and E1 .
Proof:
(1) Let A be an m x n matrix with columns i,j where i ≠ j.
(2) Let A' be the matrix A after columns i,j are interchanged.
(3) Let In be the n x n identity matrix. [See Definition 1 here for definition of the identity matrix]
(4) Let E1 be the matrix In after the columns i,j are interchanged.
(5) Then, A' = AE1. [See here for review of matrix multiplication if needed]
QED
Lemma 2:
The multiplication of any column in a matrix A by a nonzero scalar α is equivalent to a multiplication between a matrix A and E2.
Proof:
(1) Let A be an m x n matrix and let i be any column.
(2) Let A' be the matrix A after column i is multiplied by α
(3) Let In be the n x n identity matrix. [See Definition 1 here for definition of the identity matrix]
(4) Let E2 be the matrix In after the column i is multiplied by α.
(5) Then, A' = AE2. [See here for review of matrix multiplication if needed]
QED
Lemma 3:
The addition of a scalar multiple α of some column i to another column j in the matrix A is equivalent to a multiplication between a matrix A and E3.
Proof:
(1) Let A be an m x n matrix with columns i,j where i ≠ j.
(2) Let A' be the matrix A after a scalar multiple α of the column i is added to the column j.
(3) Let In be the n x n identity matrix. [See Definition 1 here for definition of the identity matrix]
(4) Let E3 be the matrix In where position i in column j is replaced by α instead of 0.
(5) Then, A' = AE3. [See here for review of matrix multiplication if needed]
QED
Lemma 4: E1, E2, and E3 in the above lemmas are all invertible.
Proof:
(1) (E1)-1 = E1
(2) (E2)-1 = E2 with α replaced by 1/α.
(3) (E3)-1 = E3 with α replaced by -α.
QED
Theorem 5: Implication of Column Equivalence
A is column equivalent to B if and only if B = AP where P is the product of elementary matrices P = Ea*Eb*...*Ez and P is invertible.
Proof:
(1) Assume that A is column equivalent to B. [See Definition 5 above for definition of column equivalence]
(2) Then A can be derived from B using n elementary operations
(3) Let A0 = A.
(4) Let A1 = A0 after the first elementary operation. Using Lemma 1, Lemma 2, or Lemma 3, we know that there exists Ea such that A1=A0Ea
(5) Let A2 = A1 after the second elementary operation. Again, it is clear that there exists Eb such that A2 = A1Eb.
(6) Using substitution, we have:
A2 = AEbEa
(7) We can continue in this way for each of the remaining elementary operations until we get:
An = AEz*...*Ea
(8) Assume that B = AP where P is the product of elementary matrices P = Ea*Eb*...*Ez
(9) Clearly, each of the Ei is equivalent to an elementary operation and we can see that B consists of a series of elementary operations since we have:
B = ([A...Ec]Eb)Ea)
(10) So, by definition of column equivalence (see Definition 4 above), we can conclude that A is column equivalent to B.
(11) We know that P is invertible since each Ei is invertible (see Lemma 4 above) and therefore any product of these elements is also invertible (see Lemma 3, here)
QED
References
- Hans Schneider, George Philip Barker, Matrices and Linear Algebra
, 1989.
No comments :
Post a Comment