Change of Basis - Matrix Algebra, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Change of basis

Consider an n x n matrix A and think of it as the standard representation of a transformation T_A : Rⁿ → Rⁿ. If we pick a different basis {v₁,... v_n} of Rⁿ, what matrix B represents T_A with respect to that new basis?

Write V = [v₁ v₂ ... v_n] and consider the diagram

which says the new matrix is B = V ^-1AV .

Remark: This is an instance of the more general change of coordinates formula. Start with a linear transformation T : V → W . Let A be the "old" basis of V and  the "new" basis. Let B be the "old" basis of  the "new" basis. Then the diagram 

gives the change of coordinates formula

Example: Let A =  viewed as a linear transformation R² → R². Find the matrix B representing the same transformation with respect to the basis 

Write V = [v₁ v₂]. Then we have

What this means is that the transformation A : R² → R² is determined by

Av₁ = 4v₁ + 0v₂ = 4v₁ (first column of B)

Av₂ = 0v₁ - 1v₂ = v₂ (second column of B).

Let us check this explicitly:

Similar matrices

The change of basis formula B = V^-1AV suggests the following denition.

Denition: A matrix B is similar to a matrix A if there is an invertible matrix S such that B = S^-1AS .

In particular, A and B must be square and A; B ; S all have the same dimensions n x n. The idea is that matrices are similar if they represent the same transformation V → V up to a change of basis.

Example: In the example above, we have shown that   is similar to

Exercise: Show that similarity of matrices is an equivalence relation.

Trace of a matrix

How can we tell if matrices are similar? There is an easy necessary (though not sucient) condition.

Denition: The trace of an n x n matrix A is the sum of its diagonal entries:

Examples: 

Proposition: For any two n x n matrices A and B, we have tr(AB) = tr(BA).

Proof:

Example:

Corollary: Similar matrices have the same trace.

Proof: Homework #8.5.

Example: In the example above, we have tr 

Warning! The converse does not hold. In other words, matrices with the same trace are rarely similar.

Example: The matrices 

all have trace 0, but none of them is similar to another, since they have different ranks (respectively 0, 1, 2).

In fact, for any number a ∈ R the matrix   has trace 0. However, if we pick a number  b ≠ +a, then  is not similar to 

In chapter 4, we will learn better tools to tell whether or not two matrices are similar.