The rank of a matrix corresponds to the count of linearly independent rows or columns within it. Consequently, the rank cannot exceed the total number of rows or columns in the matrix. For instance, in the case of a 3 × 3 identity matrix, all of its rows (or columns) are linearly independent, resulting in a rank of 3.
Real World Significance
- Solvability of systems of linear equations
- Image compression and denoising in image processing
- Analysis of economic input-output models
- Portfolio optimization and risk management in finance
- Controllability and observability in control systems
- Stoichiometry in chemistry for analyzing chemical reactions
- Population modeling and ecological studies in biology
- Dimensionality reduction in machine learning algorithms
- Representation of transformations in computer graphics
What is the Rank of a Matrix?
The rank of a matrix is the maximum number of rows or columns that are linearly independent. In simpler terms, it’s the most significant number of rows or columns that provide unique information.
For example, consider a 3×3 matrix:
A=
[1 2 3
4 5 6
7 8 9]
Here, the third row is a linear combination of the first two rows (Row 3 = Row 1 + Row 2). Therefore, the rank of matrix A is 2, as only the first two rows are linearly independent.
Properties of Rank of a Matrix
Here are some key properties of the rank of a matrix:
- The rank of a matrix is always a non-negative integer.
- The rank of a matrix is less than or equal to the minimum of the number of rows and columns in the matrix.
- Adding a row or column of zeros to a matrix does not change its rank.
- The rank of a matrix is the same as the rank of its transpose.
- For a square matrix, the matrix is invertible if and only if its rank is equal to the number of rows (or columns).
- The rank of the sum of two matrices is less than or equal to the sum of their ranks.
- The rank of the product of two matrices is less than or equal to the minimum of their ranks.
- If a matrix is in echelon form or reduced row-echelon form, its rank is the number of non-zero rows.
- The rank of a matrix plus the nullity of the matrix equals the number of columns in the matrix (Rank-Nullity Theorem).
- If a matrix has full rank (equal to the minimum of its number of rows and columns), it is said to be full rank.
