The affix -igi means to cause something to be in the state described by the root, yes?

For instance, "pereigi la libron" (to destroy the book, to cause the book to be destroyed).

Do verbs with this ending always require a direct object?

In the Esperanto Wikipedia article "Himno de Ukrainio," there is a sentence which reads "La verso signifas "Ankoraŭ ne mortis Ukrainio" kaj forte memorigas pri la himno de Serbio kaj himno de najbara Pollando..."

Memorigas apparently lacks a direct object. Is this a mistake? Would it be correct to say "memorigas la himnon..." meaning "brings the anthem to memory/makes the anthem remembered"?

Sorry for posting so many questions to the forum recently, I am just beginning to learn Esperanto and while the grammar is comparably easy, I still have many uncertainties.

Good question. Verbs with -igi are always transitive, but it's perfectly fine to omit the object when it's contextually clear or indefinite. If you search the Tekstaro for "pensig", for example, you'll see a handful of such cases, such as this from Monato: "li kapablis per originalaj pensoj pensigi."

This applies to other transitive verbs as well; for example, "Ni komencu!" is a common phrase, but you'll note that komenci is transitive. No problem - it's implied that you're starting something.

Speaking of a serial killer a detective might say 'he will kill again'. In this case the victim is unknown (the detective is not clairvoyant).

Li certe mortigos denove.

You are having a telephone conversation with someone and you hear a munching noise. You would say 'are you eating'

Ĉu vi manĝas?

I may be overlooking something, but I would guess that not including an object with a verb made transitive with -igi is much the same as when using a verb that is natively transitive.

Eltwish:Good question. Verbs with -igi are always transitive,

*Intransitive verbs with -ig- are always transitive, ĉu ne?

I'm under the impression that the transitivity of a verb with ig/iĝ is based on the transivity of the underlying root word.

From Being Colloquial in Esperanto (12.4):

"In Esperanto the vast majority of verbs are either transitive or intransitive, but not both.* Thus the Esperanto verb boli is cognate with the English “boil,” but boli is always intransitive. It means “boil” in the sense of “boil and bubble.” One cannot boli anything. In Esperanto we can say La akvo bolas. = “The water is boiling.” But we need a different verb to say “I am boiling the water.”

Sometimes such a verb already exists. But often it must be created. This can be done by use of the suffixes -ig- and -iĝ-.

The basic device for turning an intransitive Esperanto verb root into a transitive one is the suffix -ig-. ...

The basic device for turning a transitive verb root into an intransitive verb is the suffix -iĝ-. ...

But by adding -iĝ- we can make intransitive verbs that do not require direct objects: "

ETA: And, of course, after posting I finished the thought: "Intransitive verbs with -ig- are always transitive" and transitive verbs stay transitive.

johnnotegbert: Would it be correct to say "memorigas la himnon..." meaning "brings the anthem to memory/makes the anthem remembered"?

The Detailed Grammar (one of the pages of Lernu!) says in its section 27.17:
"You can also proceed from a transitive verb. The object of that type of IG-verb is either the subject of the simple verb, or the object of the simple verb:
kompreni → komprenigi = to make someone understand
Mi finfine sukcesis komprenigi lin, ke li devas iri hejmen. - I finally managed to make him understand that he had to go home.
Mi ne povis komprenigi la ŝercon al li. - I couldn't make him understand the joke."

So it seems as your suggestion is OK. Another way would be to say: "memorigas cxiujn pri la himno/makes everybody remember the antemn"

Thank you for all the responses!