Definition: An elementary matrix is a matrix obtained by performing a single elementary row operation on the identity matrix. There are three types, corresponding to the three types of row operations:
Row swap $E_{swap}$: swap rows $i$ and $j$ of $I$
Row scaling $E_{scale}$: multiply row $i$ of $I$ by nonzero scalar $c$
Row replacement $E_{replace}$: add $c$ times row $j$ to row $i$ of $I$
Key property: Left-multiplying a matrix $A$ by an elementary matrix performs the corresponding row operation on $A$:
$$EA = \text{result of applying the row operation to } A$$
Properties:
Every elementary matrix is invertible
The inverse of an elementary matrix is also elementary
Any invertible matrix can be expressed as a product of elementary matrices
Elementary matrices are fundamental in understanding Gaussian elimination and matrix factorization.
**Definition:** An **elementary matrix** is a matrix obtained by performing a single elementary row operation on the identity matrix. There are three types, corresponding to the three types of row operations:
1. **Row swap** $E_{swap}$: swap rows $i$ and $j$ of $I$
2. **Row scaling** $E_{scale}$: multiply row $i$ of $I$ by nonzero scalar $c$
3. **Row replacement** $E_{replace}$: add $c$ times row $j$ to row $i$ of $I$
**Key property:** Left-multiplying a matrix $A$ by an elementary matrix performs the corresponding row operation on $A$:
$$EA = \text{result of applying the row operation to } A$$
**Properties:**
- Every elementary matrix is invertible
- The inverse of an elementary matrix is also elementary
- Any invertible matrix can be expressed as a product of elementary matrices
Elementary matrices are fundamental in understanding Gaussian elimination and matrix factorization.