By Kardi Teknomo, PhD .
LinearAlgebra

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Matrix Rank

Recalled in the previous topics when we have a set of basis vectors , we can concatenate these vectors into a matrix. Using this matrix, we can also change the basis or the coordinate system. In this page, you will learn more about that augmented matrix.

Suppose we have a matrix Matrix Rank size Matrix Rank by Matrix Rank . We can imagine this matrix is actually a vertical concatenation of row vectors. A set of row vectors that form a matrix is called row space of the matrix.
Matrix Rank

Similarly, we can imagine the matrix Matrix Rank is actually a horizontal concatenation of column vectors. A set of column vectors that form a matrix is called column space of the matrix.
Matrix Rank

The dimension of row space of a matrix is equal to the dimension of the column space of that matrix. The common dimension of the row space and column space of matrix Matrix Rank is called the rank of matrix Matrix Rank , denoted by Matrix Rank . The rank of Matrix Rank is the number of linearly independent vectors and equal to the dimension of space spanned by those vectors. In other words, rank of Matrix Rank is the number of basis vectors we can form from matrix Matrix Rank .

Performing elementary row operation does not change the row space. To compute the rank of a matrix, we reduce the matrix into reduced row echelon form (RREF) and non-zero rows of the RREF matrix will form the basis for the row space.

The interactive program below produces matrix rank. The computation is based on numerical method of Singular Value Decomposition (SVD). The input is rectangular matrix. Random Example button will provide you with many examples.

Properties

Some important properties of matrix rank are:

  • Transpose operation does not change the rank of a matrix Matrix Rank .
  • Multiplication of a matrix with its transpose does not change the rank of the matrix Matrix Rank .
  • Rank of a matrix size Matrix Rank by Matrix Rank is always less or equal to the minimum size of the matrix. A matrix is said to be of full rank when Matrix Rank .
  • Let matrix Matrix Rank and Matrix Rank both are matrices of the same size Matrix Rank by Matrix Rank , not necessarily of full rank, then we have the following
    o Rank of matrix addition : Matrix Rank
    o Rank of matrix subtraction : Matrix Rank
  • Let matrix Matrix Rank and Matrix Rank are not necessarily the same size, then we have the following
    o Rank of matrix product : Matrix Rank
    o Rank of matrix product compare to summation of rank: Matrix Rank
  • The dimension of the range and the null space of a matrix are related through fundamental relationship Matrix Rank .
  • A linear system Matrix Rank has a solution if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix Matrix Rank . When non-homogeneous system is not full rank or when the rank of the matrix coefficients is less than the rank of the augmented matrix coefficients and the vector constants Matrix Rank then the system is usually inconsistent with no possible solution. It is still possible to find the approximate solution using generalized inverse Matrix Rank .
  • A matrix Matrix Rank has a left inverse Matrix Rank if and only if its rank equals its number of columns Matrix Rank . If matrix Matrix Rank has more columns than rows ( Matrix Rank ), it cannot have a left inverse.
  • A matrix Matrix Rank has a right inverse Matrix Rank if and only if its rank equals its number of rows Matrix Rank . If matrix Matrix Rank has more rows than columns ( Matrix Rank ), it cannot have a right inverse.
  • A matrix Matrix Rank has an inverse Matrix Rank ( non-singular ) if and only if it is a square matrix and the rank of the matrix is full, that is the rank equals to the number of rows (or columns) Matrix Rank . If matrix Matrix Rank is not square ( Matrix Rank ), it cannot have a two-sided inverse. Equivalently, the determinant of the matrix is non-zero Matrix Rank .

See also : rank through RREF , matrix range , matrix nullity and null space , solving system linear equations

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