Matrices and Linear Transformations by Charles G. Cullen

Author:Charles G. Cullen
Language: eng
Format: epub
Publisher: Dover Publications
Published: 1972-03-09T16:00:00+00:00

Now Crdβ(α) ≠ 0 because α ≠ 0, so it follows that λ is also a characteristic value of A and that the associated characteristic vector of A is Crdβ(α). The converse was established before Definition 4.7.

Theorem 4.11 says that characteristic values depend on the transformation and not on the basis chosen to represent it. From Theorems 4.10 and 4.11, we now have the following

Corollary Similar matrices have the same characteristic values.

If, as in Eq. (4.8), A is similar to a diagonal matrix, then we see from Eq. (4.9) that the columns of P are all characteristic vectors of A. Since P is nonsingular, the columns of P form a basis for n × 1 consisting entirely of characteristic vectors. Conversely, if such a basis exists, then it is clear that A is similar to a diagonal matrix. We have, therefore, proved Theorem 4.12.

Theorem 4.12 The n × n matrix A is similar to a diagonal matrix if and only if there exists a basis for n × 1 consisting of characteristic vectors of A.

We can restate Theorem 4.12 as a result about linear operators as follows.

Theorem 4.13 Let be an n-dimensional vector space over . The operator τ ∈ (, ) can be represented by a diagonal matrix if and only if there is a basis for consisting of characteristic vectors of τ.

It is important to observe that characteristic vectors are not unique. If α is a characteristic vector of τ ∈ (, ) [where τ(α) = λα], then it follows directly from the linearity of τ that,

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