full rank matrix meaning

A matrix is said to be full rank when its rank is equal to the smallest dimension of the matrix. The rank of a matrix is the maximum number of linearly independent column or row vectors in the matrix. Here’s a more detailed explanation:

Definitions

1. Matrix Dimensions:

2. For a matrix ( A ) of size ( m times n ) (where ( m ) is the number of rows and ( n ) is the number of columns), its rank can be at most ( min(m, n) ).

3. Rank:

4. The rank of a matrix is the dimension of the vector space generated by its rows or columns. It indicates the number of linearly independent vectors present.

Full Rank Conditions

• For a matrix to be considered full rank:
• If ( m leq n ) (more rows than columns or equal), the matrix is full rank if its rank is ( m ). This means all rows are linearly independent.
• If ( m > n ) (more columns than rows), the matrix is full rank if its rank is ( n ). This means all columns are linearly independent.

Key Points

1. Square Matrices:

2. If the matrix is square (i.e., ( m = n )), it is full rank if its rank is ( n ). A square matrix that is full rank is also invertible (non-singular).

3. Rectangular Matrices:

4. For rectangular matrices, being full rank means that the maximum possible number of rows or columns (whichever is smaller) are linearly independent.

5. Implications of Full Rank:

6. A full rank matrix has a unique solution for the equation ( Ax = b ) for every vector ( b ) in the appropriate dimension.
7. In the context of linear transformations, a full rank matrix represents a transformation that does not collapse any dimensions (i.e., it preserves the dimensionality of the input space to the output space).

Examples

1. Full Rank Matrix:
   [
   A = begin{pmatrix}
   1 & 2
   3 & 4
   end{pmatrix}
   ]

2. This ( 2 times 2 ) matrix has rank 2 (the maximum possible for its size), so it is full rank.

3. Not Full Rank Matrix:
   [
   B = begin{pmatrix}
   1 & 2
   2 & 4
   end{pmatrix}
   ]

4. This matrix has rank 1 because the second row is a multiple of the first. Thus, it is not full rank as it does not have 2 linearly independent rows.

Conclusion

In summary, a full rank matrix is one where the rank is maximized based on its dimensions, indicating that all rows or columns are linearly independent. This property is crucial in various applications, including solving linear systems, determining the invertibility of matrices, and understanding transformations in linear algebra.