Canonical Forms of Matrices

Canonical forms of Matrices is the method of arranging a Matrix in a standard and organised format. Another name for this form is the normalization or standardization of matrices. It is the way which gives not only unique but also simplified representation of the elements in matrices. There are various types as Echelon form and Triangular form.. Let us discuss about them-

 

Types of canonical forms

There are various types of canonical forms, among which here we are discussing

  • Jordan Form
  • Diagonal Form
  • Triangular Form

Jordan Form :

It is the transformation which is possible for every matrix. In this form, the given matrix is divided into small Jordan blocks, in such a way that the eigen value is at diagonal position and unity (one) is at the super diagonal. Here, Super diagonal position is nothing but the position after the diagonal.
This form can be understood by the example below-

In above example, the symbol λ shows the Eigen values, whereas, the element at super diagonal position is 1.

Diagonal (Echelon) Form:

It is the canonicalrm of the matrix on which we transform into the diagonal matrix. It is nothing but a matrix with all the diagonal entries are non zero and non diagonal entries are zero. In this form the leading element of each row is one, whatever maybe the remaining elements

Row Echelon form

Triangular Form:

It is a special kind of row Echelon form because, in this form, we transfer the matrix into a triangular form that is,either upper or lower triangular form,such that the leading elements (which are diagonal elements also) in each row are one(1) whereas all the entries (either upper Or lower triangular) are zero.Which in turn gives us it’s canonical form.

upper triangular matrix lower triangular

All the above are nothing but canonical forms of the matrices. We’ll discuss about them in details further …