Understand various matrix operations, matrix decompositions, factorization and related operations
Rank of a Matrix
Theory:
Definition: The rank of a matrix is defined as the maximum number of linearly independent rows (or columns) in the matrix. It can also be seen as the dimension of the row space or column space of the matrix.
Let’s consider the matrix:
A =
\( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
Convert it to REF:
REF(A) =
\( \begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & 0 & 0 \end{bmatrix} \)
This matrix has 2 non-zero rows, so:
Rank(A) = 2
Example 2: Zero Matrix (No Rank)
B =
\( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
There are no non-zero rows.
Rank(B) = 0