The rank of a matrix is a fundamental concept in linear algebra that refers to the maximum number of linearly independent rows or columns in a matrix. It essentially tells us the dimensionality of the space spanned by the rows or columns, and it plays an important role in solving systems of linear equations.
Definition:
The rank of a matrix is the number of linearly independent rows or columns in the matrix. It indicates the maximum number of linearly independent vectors that can be formed from the rows or columns.
Properties:
Row and Column Rank: The rank of a matrix is the same whether we consider its rows or columns, i.e., the row rank and the column rank are equal.
Rank Bound: The rank of a matrix is always less than or equal to the smaller of the number of rows or columns.
Full Rank: If the rank of a matrix equals the smaller of its number of rows or columns, it is called a full-rank matrix.
Zero Rank: A matrix with all zero elements has a rank of 0.
Rank and Solutions: The rank of a matrix is crucial in determining the number of solutions to a system of linear equations. If the rank is less than the number of variables, there may be infinitely many solutions or none at all.
Formula:
While there isn’t a simple closed-form formula for the rank, it can be determined using several methods:
Row Echelon Form: The rank is equal to the number of non-zero rows in the row echelon form (or reduced row echelon form) of the matrix.
Determinants: If a square matrix has a non-zero determinant, its rank is equal to the size of the matrix. If the determinant is zero, the rank is less than the size.
In practice, finding the rank of a matrix often involves using Gaussian elimination to reduce it to row echelon form or applying the singular value decomposition.