Row equivalence is a relation between matrices of the same dimensions in linear algebra. Two matrices $A$ and $B$ of size $m \times n$ are said to be row‑equivalent if there exists a finite sequence of elementary row operations that transforms $A$ into $B$. Elementary row operations consist of:
- Swapping two rows;
- Multiplying a row by a non‑zero scalar;
- Adding a scalar multiple of one row to another row.
Formally, $A$ and $B$ are row‑equivalent if there exists an invertible $m \times m$ matrix $E$ (a product of elementary matrices) such that
$$ B = EA . $$
Key Properties
- Equivalence Relation: Row equivalence is reflexive, symmetric, and transitive, thereby partitioning the set of all $m \times n$ matrices into equivalence classes.
- Preservation of Rank: All matrices within a row‑equivalence class have the same rank. Conversely, matrices of the same size and rank are not necessarily row‑equivalent; additional constraints on the row space are required.
- Row‑Reduced Echelon Form (RREF): Each row‑equivalence class contains a unique matrix in reduced row‑echelon form. This canonical representative is often used to determine the class to which a given matrix belongs.
- Solution Sets of Linear Systems: If the augmented matrices of two linear systems are row‑equivalent, the systems have the same solution set. This underlies the method of Gaussian elimination.
Applications
- Solving Linear Systems: Row operations are applied to transform a system’s coefficient matrix (or augmented matrix) into a simpler, equivalent form from which solutions can be read directly.
- Computing Matrix Rank: By reducing a matrix to RREF, the number of non‑zero rows yields the rank, which is invariant under row equivalence.
- Determining Linear Independence: Row‑equivalence aids in assessing whether a set of vectors (as rows) is linearly independent.
Related Concepts
- Column Equivalence: Analogous to row equivalence, but based on column operations; it preserves column space and column rank.
- Matrix Equivalence: Two matrices $A$ and $B$ are equivalent if there exist invertible matrices $P$ and $Q$ such that $B = PAQ$; this is a stricter relation than row equivalence.
- Elementary Matrices: Matrices that represent single elementary row operations; multiplying a matrix by an elementary matrix on the left effects the corresponding row operation.
Historical Note
The systematic use of elementary row operations and the concept of row equivalence originated in the development of the theory of linear equations in the 19th century, particularly through the work of Carl Friedrich Gauss, whose method of elimination laid the foundation for modern linear algebra.