Matrix Similarity

Similarity to a Diagonal Matrix

Let U be an n-dimensional vector space with a given basis G = {u₁, u₂ , ... , u_n}. Also, let V be an m-dimensional vector space with a given basis H= {v₁, v₂, ... ,v_m}. And let T: U --> V be a given linear transformation from U into V. Every vector x in U has a unique representation as a linear combination of the basis vectors in G, i.e.,

  x = x₁u₁ + x₂u₂ + ... + x_nu_n

Since T is a linear transformation, we have

  T(x) = T(x₁u₁ + x₂u₂ + ... + x_nu_n) = x₁T(u₁) + x₂T(u₂) + ... + x_nT(u_n)

Because H is a basis for V, each of the vectors T(u_i) has a unique representation as a linear combination of the basis vectors H for V.

  T(u_i) = a_1iv₁ + a_2iv₂ + ... + a_miv_m 

Define the matrix representation of the linear transformation T: U --> V with respect to the bases G for U and H for V to be the matrix 

  A = [ [T(u₁)]_H : [T(u₂)]_H : ... : [T(u_n)]_H ]

  A =  [ T ]^G_H

Transformation T(x) = [ T ]^G_H [x]_G = A[x]_G 

Let T:U --> U be a linear operator on the finite-dimensional vector space U. Let U have bases G and  H. Let A = [T]_G be the matrix representation of T relative to the G-basis and let B = [T]_H be the matrix representation of T relative to the H-basis. Then there exist a nonsingular matrix S such that A = SBS^-1


The matrix S is the change of basis matrix from the basis H to the basis G. It is the matrix representation of the identity linear operation I(u) = u on U, i.e. S = [ I ]^H_G . Since S represents the identity transformation, it is clearly invertible. So S^-1 = [ I ]^G_H represents the change of basis from G-basis to H-basis.

The relationship A = SBS^-1 between two different representations A and B for the same linear transformation T is used in the similarity of matrix. An nxn matrix A is said to be similar to the nxn matrix B if there exists an nxn nonsingular matrix S so that A = SBS^-1. 

Similar matrices have the same eigenvalues. An nxn Matrix A is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. Furthermore, in this case, the diagonal elements of D are the eigenvalues of A as well as those of D. Eigenvectors corresponding to different eigenvalues are always linearly independent of each other. 

An nxn matrix A is said to be diagonalizable if there exists an invertible matrix P such that the matrix P^-1AP = D is a diagonal matrix. If A is diagonalized, then it must have n linearly independent eigenvectors. Not every nxn matrix A is similar to a diagonal matrix, i.e. not every nxn matrix can be diagonalized. 

Similarity Classes

An equivalence relation on a set of objects (in this case matrices) is any relation that is reflexive, symmetric, and transitive. 

Similarity of matrices has the following important features:

1. Reflectivity : Every nxn matrix A is similar to itself.


2. Symmetry : If A is similar to B, then B is similar to A.


3. Transitivity : If A is similar to B and B is similar to C, then A is similar to C.

All matrices similar to a given nxn matrix A belong to a class called similarity class. Every nxn matrix belongs to exactly one similarity class, and no two distinct similarity classes have any elements (i.e., matrices) in common.

If K = λI is a scalar matrix, then the only matrix similar to K is K itself. Other matrices have many matrices in their individual similarity class. Each linear transformation T gives to a similarity class of matrix representation of T. Real symmetric matrices and their complex counterparts, Hermitian matrices are always diagonalizable. Similar matrices always have the same eigenvalues, and corresponding eigenspaces (but not conversely).  The method for finding the "simplest" member of the similarity class containing a given matrix is the "canonical representative" of the associated transformation (connection between eigenvalues and a diagonal "canonical similarity form" for a given matrix).