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It is equivalent to "A complex number λ is called an eigenvalue of T if Tx = λx for some nonzero x"

by zodiac 9y ago

pacala 9y ago
Thanks! Looking up wikipedia, "In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness". From that definition, "T −λI is not injective" means that there exists 2 vectors, y and z, such that (T −λI)y = (T −λI)z. Giving T(y - z) = λI(y - z). Via x = (y - z), I get to your (usual) definition. The question I have is _why_ use the injective-based definition instead of the usual well-known one? Is there some further insight down the road?
- Koshkin 9y ago
  The merit in using the purely functional language is in that it opens the door for application of powerful methods and concepts of Category Theory. (Note also that even the very notion of a function being injective can be defined in a purely functional manner, i.e. without any reference to sets or their elements.)
  - zodiac 9y ago
    Linear Algebra is a great place to get exposed to some category-theoretical concepts (maybe not when first learning linear algebra, though). My linear algebra prof used the category-theoretical definitions of subspace and quotient space (eg S is a subspace of L if there exists an injective linear map from S to L, a quotient space if there is an injective linear map) and tensor product (universal property)
- zodiac 9y ago
  With the injectivity-based definition, you can ask "how injective is it?" in various ways (eg, what is the dimension of the nullspace of T - λI; the Ax = λx definition is rather binary and corresponds to simply "T - λI has nullity > 0"). This is what is meant by "eigenvalue multiplicity" (there are actually at least two different inequivalent ways to measure this, geometric multiplicity and algebraic multiplicity). Also for finite-dimensional vector space a lot of concepts are equivalent, eg "a linear operator L is invertible <=> L is injective", which are not in infinite-dimensional vector spaces (eg https://math.stackexchange.com/a/2447563/21437 https://math.stackexchange.com/a/2447563/21437). Studying T - λI turns out to be more useful in this case (eg https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis) https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(fun...)

It is equivalent to "A complex number λ is called an eigenvalue of T if Tx = λx for some nonzero x"

by zodiac 9y ago

It is equivalent to "A complex number λ is called an eigenvalue of T if Tx = λx for some nonzero x"

Thanks! Looking up wikipedia, "In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness". From that definition, "T −λI is not injective" means that there exists 2 vectors, y and z, such that (T −λI)y = (T −λI)z. Giving T(y - z) = λI(y - z). Via x = (y - z), I get to your (usual) definition. The question I have is _why_ use the injective-based definition instead of the usual well-known one? Is there some further insight down the road?

Koshkin 9y ago
The merit in using the purely functional language is in that it opens the door for application of powerful methods and concepts of Category Theory. (Note also that even the very notion of a function being injective can be defined in a purely functional manner, i.e. without any reference to sets or their elements.)
- zodiac 9y ago
  Linear Algebra is a great place to get exposed to some category-theoretical concepts (maybe not when first learning linear algebra, though). My linear algebra prof used the category-theoretical definitions of subspace and quotient space (eg S is a subspace of L if there exists an injective linear map from S to L, a quotient space if there is an injective linear map) and tensor product (universal property)
zodiac 9y ago
With the injectivity-based definition, you can ask "how injective is it?" in various ways (eg, what is the dimension of the nullspace of T - λI; the Ax = λx definition is rather binary and corresponds to simply "T - λI has nullity > 0"). This is what is meant by "eigenvalue multiplicity" (there are actually at least two different inequivalent ways to measure this, geometric multiplicity and algebraic multiplicity). Also for finite-dimensional vector space a lot of concepts are equivalent, eg "a linear operator L is invertible <=> L is injective", which are not in infinite-dimensional vector spaces (eg https://math.stackexchange.com/a/2447563/21437 https://math.stackexchange.com/a/2447563/21437). Studying T - λI turns out to be more useful in this case (eg https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis) https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(fun...)