Introduction to rank and eigenvalues of a matrix

A matrix can be defined as the arrangement of digits in a rectangular way. It involves rows and columns. The order of a given matrix is the number of rows and columns present in the given matrix. The number of linearly independent rows/columns that exist in a matrix defines the rank of matrix. The rank is represented by rank (A) or rk (A). The rank of a matrix is one of the most important characteristics of a matrix. The image of the matrix is the possible linear combination of the column vectors.

The same holds for row space. One of the fundamental results states that the rank of the column and the row are the same. It can be calculated by reducing the given matrix into row echelon form through elementary row operations. The nonzero rows of the reduced echelon form matrix give the rank of matrix. The row operations don’t alter the row space. A few properties of the rank of a matrix are as follows.

  • The rank of a matrix is an integer that is non-negative.
  • A matrix consisting of rank min (m, n) is said to have full rank, else the matrix is rank deficient.
  • A zero matrix can have its rank as 0.
  • The given square matrix A is said to be invertible when it has full rank.
  • Let B be a matrix of order n × k, then rank (AB) less than or equal to the min (rank (A), rank (B)).
  • Let B be a matrix of order n × k and rank n, then rank (AB) equals rank (A).
  • Let C be a matrix of order l × m and rank m, then rank (CA) equals the rank (A).
  • The rank of a matrix + the number of vectors that exist in the null space of the matrix (nullity of the matrix) equals the number of columns of the matrix.

Eigenvalues of a matrix

The group of scalars related to the linear system of equations can be defined as the eigenvalues. The eigenvalues of a matrix A are the λ values satisfying the equation |A – λI| = 0, where I is the identity matrix. The other names for eigenvalues are characteristic roots or characteristic values. Eigen decomposition refers to the breaking of the given square matrix into the eigenvalues and eigenvectors of the given matrix. The properties of eigenvalues are listed below.

  • The value of the determinant is the product of the eigenvalues.
  • If the eigenvalue is not 0, then the matrix is invertible.
  • The sum of the diagonal elements defines the trace of a matrix as also the sum of the eigenvalues.
  • Each eigenvalue has an absolute value if matrix A is unitary.
  • Each eigenvalue is real if the given matrix A = conjugate transpose or if A is Hermitian. The same holds good for symmetric real matrices.

This page explains the rank and eigenvalues of a matrix, their definitions, properties. For more information and solved problems on different topics of maths, please visit BYJU’S website.

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