The diagonal of a square matrix consists of entries from the upper-left to the bottom-right

Furthermore, because eigenvalues of a map are the entries of the diagonal of its upper-triangular matrix, and this is technically an upper-triangular matrix, the entries on the diagonal are exactly the eigenvalues of the Linear Map.

properties of diagonal matrices

Suppose \(V\) is finite-dimensional, and \(T \in \mathcal{L}(V)\); and let \(\lambda_{1}, … \lambda_{m}\) be distinct eigenvalues of \(T\). Then, the following are equivalent:

see Diagonalization Arguments

A human Dialogue is a human to human interaction.

turn

each contributino to a conversation is called a “turn”, which contains a sentence, multiple sentences, or a single word

turn-taking

• when to take the floor
• who takes the floor
• what happens during interruptions?

barge-in

barge-in is the property to allow the user to interrupt the system

end-pointing

deciding when a human has stopped talking, compute, etc.

speech-act

each turn is actually an “action” performed by the user

Dialogue State Architecture uses dialogue acts instead of simple frame filling to perform generation; used currently more in research.

• NLU: slot fillers to extract user’s utterance, using ML
• Dialogue State Tracker: maintains current state of dialogue
• Dialogue policy: decides what to do next (think GUS’ policy: ask, fill, respond)—but nowaday we have more complex dynamics
• NLG: respond

dialogue acts

dialogue acts combines speech-acts with underlying states

slot filing

we typically do this with BIO Tagging with a BERT just like NER Tagging, but we tag for frame slots.

An