Lecture 22: Introductory Linear Algebra - Elementary Matrices

Introduction to Elementary Matrices

• Elementary matrices are to matrices what prime numbers are to integers.
• Almost every matrix can be expressed as a product of elementary matrices.
• Useful for proving properties of general matrices by first proving for elementary matrices.

Relationship with Row Operations

• Elementary matrices correspond one-to-one with row operations.
• They can be used to perform row operations such as:
  • Row swaps
  • Row additions
  • Scalar row multiplications

Examples of Elementary Matrices

• Given a matrix, performing a row swap can be achieved by multiplying with a matrix designed to swap rows.
• Example: Row operation row 3 - 3*row 1 can be performed by multiplying with a corresponding elementary matrix.
• Scalar operations like multiplying a row by a constant can also be achieved using elementary matrices.

Constructing Elementary Matrices

• Start from the identity matrix and perform the desired row operation.
• Example 1: Swapping rows 1 and 2 in the identity matrix gives the corresponding elementary matrix.
• Example 2: Performing row 3 - 3*row 1 starting from identity gives another elementary matrix.

Definition of Elementary Matrices

• A matrix is elementary if it can be obtained from the identity matrix via a single row operation.

General Structure of Elementary Matrices

• Swap row operation: Looks like identity with two rows swapped.
• Addition row operation: Identity matrix with an extra entry.
• Scalar multiplication row operation: Identity with a different scalar on the diagonal.

Application in Row Reduction

• Multiplying a matrix by elementary matrices reduces it to its reduced row echelon form (RREF).
• The sequence of operations to get to RREF can be expressed as a product of elementary matrices.

Method to Derive Matrix E

• Augment matrix A with an identity matrix on the right.
• Perform row operations to achieve RREF on the left.
• The resulting matrix on the right is the product of elementary matrices needed to achieve RREF.

Block Matrix Multiplication

• Demonstrates how multiplication by elementary matrices corresponds to row operations.
• Using block matrix multiplication helps in theoretically proving the relation between matrices and row operations.

Theorem: Elementary Matrix Multiplication

• If an augmented matrix (A|I) is reduced to (R|E), then R = E * A.
• Particularly useful when R is the reduced row echelon form of A.

Conclusion

• Theoretical significance: Elementary matrices and their operation sequences are vital for proofs and understanding matrix properties.
• Next class will focus on invertible matrices.