Echelon matrices
Definition: A matrix is in row echelon form if:
- All nonzero rows are above any zero rows
- The leading entry (pivot) of each nonzero row is to the right of the leading entry of the row above it
- All entries below each pivot are zero
A matrix is in reduced row echelon form (RREF) if additionally:
4. Each pivot is 1
5. Each pivot is the only nonzero entry in its column
Example (row echelon form):
$$\begin{pmatrix} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
Example (RREF):
$$\begin{pmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
Echelon form is obtained through Gaussian elimination. The number of nonzero rows equals the rank of the matrix. Every matrix has a unique RREF.
Parents
Children
- Example of putting a matrix in echelon form
- Example of putting a matrix in echelon form and identifying the pivot columns
- Definition of (echelon matrix/matrix in (row) echelon form)
- Gauss-Jordan procedure to put a matrix into reduced row echelon form
- Example of (echelon matrix/matrix in (row) echelon form)
- Definition of (row) echelon form of a matrix
- Definition of matrix in reduced row echelon form
- Definition of reduced row echelon form of a matrix
- Definition of row reduce a matrix
- Example of row reducing a 3-by-3 matrix
- Example of row reducing a 4-by-4 matrix
- Every matrix is row-equivalent to a matrix in reduced row echelon form.
- Every matrix is row-equivalent to only one matrix in reduced row echelon form.
- Definition of extended reduced row echelon form of a matrix
- Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrix
- Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrix
- Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrix