invariant subspaces are a property of operators; it is a subspace for which the operator in question on the overall space is also an operator of the subspace.

constituents

• an operator \(T \in \mathcal{L}(V)\)
• a subspace \(U \subset V\)

requirements

\(U\) is considered invariant on \(T\) if \(u \in U \implies Tu \in U\)

(i.e. \(U\) is invariant under \(T\) if \(T |_{U}\) is an operator on \(U\))

additional information

nontrivial invariant subspace

(i.e. eigenstuff)

A proof is not given yet, but \(T \in \mathcal{L}(V)\) has an invariant subspace that’s not \(V\) nor \(\{0\}\) if \(\dim V > 1\) for complex number vector spaces and \(\dim V > 2\) for real number vector spaces.

the inverse is the the opposite of an operation. As in, if you apply the inverse of an operation to the result of applying the original with the same operation it will cancel it.

That is,

\begin{equation} A * B * B^{-1} = A \end{equation}

\(B^{-1}\) is then the inverse of \(B\) for the \(*\) operation. This is operation dependent.

1. Generate a uniform number between 0 to 1.
2. Get the inverse of the standard normal density function at that value (let number be \(y\), find the \(x\) such that \(\phi(y) = x\))
3. return \(x\)

A Linear Map is invertable if it can be undone. It is called a nonsingular matrix

constituents

A linear map \(T \in \mathcal{L}(V,W)\)

requirements

A Linear Map \(T \in \mathcal{L}(V,W)\) is called invertable if \(\exists T^{-1} \in \mathcal{L}(W,V): T^{-1}T=I \in \mathcal{L}(V), TT^{-1} = I \in \mathcal{L}(W)\).

“a map is invertable if there is an inverse”: that combining the commutable inverse and itself will result in the identity map.

additional information

matrix invertability

Matrices whose determinants are not \(0\) (i.e. it is invertable) is called “nonsingular matrix”. If it doesn’t have an inverse, it is called a singular matrix.

For each term \(t\), let’s store all the documents containing \(t\). We identify each doc by DocID.

postings list

a “postings list” datastructure is a variable-length list which is appended to with “postings”. In this way, we can store a “posting” for every DocID with the index we encounter.

For instance, this could be a linked list.

Although: we generally want to sort our postings list by documentID for ease of indexing.