Decomposing Matrices with Elementary Operations

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Elementary matrices are like the building blocks of matrix operations. They simplify complex matrices, help us understand Gaussian elimination, and reveal important relationships between matrices.

Elementary Matrices

• What they are: Elementary matrices are special types of non-singular matrices obtained by performing a single elementary row or column operation on an identity matrix.
• Why they’re important:
  • They provide a clear, formal way to represent and perform elementary row/column operations.
  • They are fundamental to understanding Gaussian elimination and Gauss-Jordan elimination.
  • Their inverses are also elementary matrices of the same type, which means every elementary operation is reversible by another elementary operation.
  • Any invertible (non-singular) matrix can be expressed as a product of elementary matrices.
• The Big Three Types: We primarily focus on three types of elementary matrices, corresponding to the three elementary row/column operations:
  • Type I (Row/Column Swap): Obtained by interchanging two rows or columns of the identity matrix.
  • Type II (Row/Column Scaling): Obtained by multiplying a single row or column of the identity matrix by a non-zero scalar.
  • Type III (Row/Column Addition): Obtained by adding a multiple of one row/column to another row/column of the identity matrix.
• Left vs. Right Multiplication:
  • Multiplying a matrix on the left by an elementary matrix performs a row operation.
  • Multiplying a matrix on the right by an elementary matrix performs a column operation.

Equivalence and Uniqueness

• Nonsingular Matrices: Any nonsingular matrix can be expressed as a product of elementary matrices. This means you can get to any nonsingular matrix by a sequence of these basic operations.
• Equivalent Matrices: If matrix $B$ is obtained from matrix $A$ by multiplying $A$ by elementary matrices (either on the left for row operations or on the right for column operations), then $A$ and $B$ are considered equivalent matrices.
• Why equivalence matters:
  • Row Equivalence: If $A$ and $B$ are row equivalent (meaning $B$ was obtained from $A$ using only row operations), they have the same row space and the same null space (set of solutions to $Ax=0$).
  • Column Equivalence: Similarly, if $A$ and $B$ are column equivalent, they have the same column space and the same left null space (set of solutions to $x^{T}A=0$).
• Unique Forms:
  • Row Reduced Echelon Form (RREF): This is a unique form obtained by applying only row operations to a matrix. It’s super useful for solving systems of linear equations, finding the null space, and determining the rank of a matrix.
  • Rank Normal Form: This form is achieved by applying both row and column operations. It’s a highly simplified diagonal form that explicitly shows the rank of the matrix.

(Meyer, 2000, pp. 131–138)

Reference

Meyer, C. D. (2000). Matrix Analysis and Applied Linear Algebra (Vol. 71). SIAM.