Rank of a Matrix

  • Rank of a Matrix

An important characteristic of any matrix is its rank. It tells us the number of independent rows or columns of matrix.

Note that all matrices have a rank, unlike inverse of a matrix, which only a non-singular square matrix has.

Definition: A matrix has a rank {r}, if

I) At least one  minor of order {r} which is not equal to {0} and

II) Every minor of order {r+1} is {0}.

[A minor of a matrix {A} is the determinant of some smaller square matrix, formed by removing one or more rows or columns of {A}. ]

The elementary transformations of a matrix do not alter its rank. Any matrix {B}, obtained by transforming a matrix {A} is known as equivalent matrix. It is denoted by {A \sim B}.

  • Rank of a Matrix by reducing it to Echelon/ Canonical Form

A matrix is in echelon/ canonical form if

I) all nonzero rows are above any rows of all zeroes (if any), and

II) the leading coefficient (the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it. The pivot is preferably taken to be {1}.

The rank {r} is the number of non-zero rows of matrix.

  • Rank of a Matrix by reducing it to Normal Form

A normal form of a non-zero matrix {A} is a matrix in either of the following forms:

{[I_r], \ \begin {bmatrix} I_r & 0 \end {bmatrix}, \ \begin {bmatrix} I_r \\ 0 \end{bmatrix}, \ \begin {bmatrix} I_r & 0 \\ 0 & 0 \end {bmatrix}}

{I_r} is identity matrix of order {r}.

We use both {row} and {column} transformations to reduce the matrix to normal form. The rank of the matrix is then equal to {r}.

  • Obtaining 2 matrices {P} and {Q}, such that {PAQ} is in normal form

Consider a matrix {A} with {m} rows and {n} columns. Take {P} as {I_m} and {Q} as {I_n}. We can then write

{A_{m \times n} = I_{m} \times A \times I_n = PA Q}

Reduce {A} on the LHS to a normal form. Perform respective row transformations on {I_m} and column transformations on {I_n}. Once {A} is reduced to normal form, we get the rank.

Note: {P} and {Q} are not unique, and are non-singular.

  • Inverse using {PAQ} form

If {PAQ} is in normal form, {A^{-1}} is {QP}.

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