Similarity

The following two questions are closely related to those of the preceding section.

Question III. If is a linear transformation on , what is the relation between its matrix with respect to and its matrix with respect to ?

Question IV. If is a matrix, what is the relation between the linear transformations and defined, respectively, by and ?

Questions III and IV are explicit formulations of a problem we raised before: to one transformation there correspond (in different coordinate systems) many matrices (question III) and to one matrix there correspond many transformations (question IV).

Answer to question III. We have and Using the linear transformation defined in the preceding section, we may write and Comparing (2), (3), and (4), we see that Using matrix multiplication, we write this in the dangerously simple form The danger lies in the fact that three of the four matrices in (5) correspond to their linear transformations in the basis ; the fourth one –namely, the one we denoted by – corresponds to in the basis . With this understanding, however, (5) is correct. A more usual form of (5), adapted, in principle, to computing when and are known, is

Answer to question IV. To bring out the essentially geometric character of this question and its answer, we observe that and Hence is such that or, finally, There is no trouble with (7) similar to the one that caused us to make a reservation about the interpretation of (6); to find the linear transformation (not matrix) , we multiply the transformations , , and , and nothing needs to be said about coordinate systems. Compare, however, the formulas (6) and (7), and observe once more the innate perversity of mathematical symbols. This is merely another aspect of the facts already noted in Sections 37 and 38.

Two matrices and are called similar if there exists an invertible matrix satisfying (6); two linear transformations and are called similar if there exists an invertible transformation satisfying (7). In this language the answers to questions III and IV can be expressed very briefly; in both cases the answer is that the given matrices or transformations must be similar.

Having obtained the answer to question IV, we see now that there are too many subscripts in its formulation. The validity of (7) is a geometric fact quite independent of linearity, finite-dimensionality, or any other accidental property that , and may possess; the answer to question IV is also the answer to a much more general question. This geometric question, a paraphrase of the analytic formulation of question IV, is this: If transforms , and if transforms the same way, what is the relation between and ? The expression "the same way" is not so vague as it sounds; it means that if takes into, say, , then takes into . The answer is, of course, the same as before: since and (where and ), we have The situation is conveniently summed up in the following mnemonic diagram: We may go from to by using the short cut , or by going around the block; in other words . Remember that is to be applied to from right to left: first , then , then .

We have seen that the theory of changing bases is coextensive with the theory of invertible linear transformations. An invertible linear transformation is an automorphism , where by an automorphism we mean an isomorphism of a vector space with itself. (See Section: Isomorphism .) We observe that, conversely, every automorphism is an invertible linear transformation.

We hope that the relation between linear transformations and matrices is by now sufficiently clear that the reader will not object if in the sequel, when we wish to give examples of linear transformations with various properties, we content ourselves with writing down a matrix. The interpretation always to be placed on this procedure is that we have in mind the concrete vector space (or one of its generalized versions ) and the concrete basis defined by . With this understanding, a matrix defines, of course, a unique linear transformation , given by the usual formula

EXERCISES

Exercise 1. If is a linear transformation from a vector space to a vector space , then corresponding to each fixed in there exists a vector, which might as well be denoted by , in so that for all in . Prove that is a linear transformation from to . (The transformation is called the adjoint of .) Interpret and prove as many as possible among the equations Section: Adjoints , (2)-(8) for this concept of adjoint.

Exercise 2.

Prove that similarity of linear transformations on a vector space is an equivalence relation (that is, it is reflexive, symmetric, and transitive).
If is similar to a scalar , then .
If and are similar, then so also are and , and , and, in case and are invertible, and .
Generalize the concept of similarity to two transformations defined on different vector spaces. Which of the preceding results remain valid for the generalized concept?

Exercise 3.

If and are linear transformations on the same vector space and if at least one of them is invertible, then and are similar.
Does the conclusion of (a) remain valid if neither nor is invertible?

Exercise 4. If the matrix of a linear transformation on , with respect to the basis is what is the matrix of with respect to the basis ? What about the basis ?

Exercise 5. If the matrix of a linear transformation on , with respect to the basis is what is the matrix of with respect to the basis ?

Exercise 6.

The construction of a matrix associated with a linear transformation depends on two bases, not one. Indeed, if and are bases of , and if is a linear transformation on , then the matrix of with respect to and should be defined by The definition adopted in the text corresponds to the special case in which . The special case leads to the definition of similarity ( and are similar if there exist bases and such that ). The analogous relation suggested by the general case is called equivalence; and are equivalent if there exist basis pairs and such that . Prove that this notion is indeed an equivalence relation.
Two linear transformations and are equivalent if and only if there exist invertible linear transformations and such that .
If and are equivalent, then so also are and .
Does there exist a linear transformation such that is equivalent to a scalar , but ?
Do there exist linear transformations and such that and are equivalent, but and are not?
Generalize the concept of equivalence to two transformations defined on different vector spaces. Which of the preceding results remain valid for the generalized concept?

EXERCISES

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