Matrix similarity
Switching bases
The basis consisting of unit vectors is simple in that the coefficients in the representation are exactly the components of the vector
With other types of bases it is not like that: the dependence of coefficients in
(1)
on for a general basis
is not so simple.
Exercise 1. Put the basis vectors side by side, and write the vector of coefficients
as a column vector. Then (1) becomes
so that
Proof. By the basis definition, runs over
and therefore
This implies
The rest is obvious.
The explicit formula from Exercise 1 shows, in particular, that the vector of coefficients is uniquely determined by and depends linearly on The coefficients
of
in another basis
(2)
may be different from those in (1). For future applications, we need to know how the coefficients in one basis are related to those in another. Put the basis vectors side by side, and write
as a column vector.
Exercise 2. Let and
be two bases in
Then
(3)
Proof. With our notation (1) and (2) become and
Thus,
and (3) follows.
Definition 1. The matrix in (3) is called a transition matrix from
to
.
Matrix representation of a linear transformation
This topic in case of an orthonormal basis was considered earlier. Some people find the next general construction simpler.
Let be a basis and decompose
as in (1). Let
be a linear transformation. From
(5)
we see that the vectors uniquely determine
. Decompose them further as
Let us introduce a column-vector
by
Then the last equation takes the form
With the matrix
we can write one equation instead of
(6)
Combining (5), (6) and Exercise 1 we get Since
is arbitrary, the linear transformation
in the basis
can be identified with the matrix
(7)
Definition 2. The matrix that is defined by (6) or (7) is called a representation of the linear transformation
in the basis
Note two special cases: 1) if the basis is orthonormal, then
is an orthogonal matrix and 2) when we use the orthonormal basis of unit vectors,
and
Changing bases to analyze matrices
We want to see how the representations of a linear transformation in two bases are related to each other. For a basis
summarized in the matrix
we have (7). To reflect dependence of
on the basis, let us denote it
Then from (7)
Similarly, for another basis
we have
The last two equations lead to
Hence,
Here
is the transition matrix, so this can be written as
(8)
Definition 3. If there is a nonsingular matrix such that (8) is true, then the matrix
is called similar to
Depending on the choice of the bases, one matrix may be simpler than the other.
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