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Similar matrices

If two matrices A and B are similar, then they represent the same linear transformation T: V -> V, where V is any vector space, under different choices of basis. This page explains this idea and shows why similarity is equivalent to the existence of a matrix C such that AC = CB.

Linear transformations and coordinate vectors

A linear transformation T: V -> V maps elements from vector space V to other elements in V. That is, T(x) = y, where x and y are in V. Now consider two bases for V: v and w. x and y can be written as a linear combination of either v or w. The weights of the linear combination will depend on which basis we choose. For instance, we can write the R2 vector [2; 3] as 2 [1; 0] + 3 [0; 1] but also as 4 [1; 1] + (-1) [2; 1].

If we select v to be our basis, we'd find x = a v and y = c v, for some weights vectors (or coordinates relative to basis v) a and c. If we know that a is mapped to c, we immediately know that x is mapped to y. For instance, say we map [2; 3] to [0; 1]. Written in the standard basis, this means mapping 2 [1; 0] + 3 [0; 1] to 0 [1; 0] + 1 [0; 1]. We might as well say we're mapping the 2 to the 0 and the 3 to the 1. Why this is useful is of course more clear when V is less like R2. We may be mapping functions together for instance, in which case being able to work with their coordinate vectors allows a great deal of abstraction to be applied: we can handle functions, or any other vectors, just like vectors in Rn!

Because of the properties of linear transformations, the coordinate-mapping can be specified as a matrix multiplication. Since the basis vectors are mapped in a certain way, so are their linear combinations. Consider a given basis. T maps the basis vectors to other vectors in V. So, the unit coordinate vectors are mapped to some other coordinate vectors. Now construct the matrix A containing the mapped coordinate vectors. For any vector x in V, we can write x as a linear combination of the basis vectors. Since T is a linear transformation, the coordinates of T(x) are the same linear combination of the coordinate vectors in A. Thus as T maps x to T(x), A times the coordinates of x results in the coordinates of T(x).

Relating the bases, relating coordinate vectors

Since v is a basis, every vector in w can be uniquely described as a linear combination of v. In other words there exists a matrix C such that w = Cv. Now, for a given x in V, we can write x = a v and x = b w. By rewriting w, we get a v = b (C v) = (b C) v. So the coordinate vectors for the respective bases are related via a = b C.

Relating the transformation matrices

Now we can relate the transformation matrices that map the coordinate matrices. Let A and B be the transformation matrices relative to bases v and w. For any x in V, let a and b be the respective coordinate vectors. Then (Aa) v = T(x) = (Bb) w. Since a = b C (from the coordinate vector relationship) and w = C v, this implies (A b C) v = (B b) (C v). Shifting associations, we get (A b C) v = (B b C) v, and so A b C = B b C. Taking the transpose twice gets A C b = C B b, and so A C = C B.

Since all the steps going from the condition w = C v to the consequence A C = C B are bidirectional, if A C = C B, then C connects the bases for which A and B are the coordinate-transformation matrices for the same linear transformation T.