## 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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