\chapter{Multilinear algebra}


Essential for tensors.

Fundamental in differential geometry.



\section{Multilinear maps}

We have studied linear maps, and we have implicitly looked as some bilinear map; bivariate functions on two linear spaces that are linear (matrix multiplication and the inner product). Indeed there are many noteworthy multilinear maps aside from these, even if these two are the most renown. 

\begin{definition}[Multilinear map]
A \emph{mulilinear map} or \emph{multilinear transform} is a function $f : V^n \to W$
\begin{itemize}
	\item $f$ is additive in every argument
	\item $f$ is homogeneous of degree $1$ in every argument
\end{itemize}
\end{definition}

This can be though of as a 'multivariate linear map'! Inner products, the cross product, and quadratic forms are examples of multilinear maps that we are already familiar with. We will now commence a deeper study on these operations generally.

\subsection{Outer product}
\subsection{Hadamard product}
\subsection{Kronecker product}










\section{Direct sum}

\section{Tensor product}

- Constructing as a quotient space
- Constructing from the basis of the underlying linear spaces
- Constructing by a univeral property


Universal properties are a concept from category theory that may be inaccessible to us, and although we are familiar with the notion of a basis the cleanest way is by quotient spaces.

\begin{definition}[Tensor product]
Let $Z$ be a linear space with $V\times W$ as its (hamel) basis.
\[\mathbf{v} \in V \]
\[\mathbf{w} \in W \]
\[c \in F\]
\[ c(\mathbf{v} * \mathbf{w}) = c\mathbf{v} * \mathbf{w} = \mathbf{v} * c\mathbf{w} \]
\[ (\mathbf{v}_1+\mathbf{v}_2 ) * \mathbf{w} = \mathbf{v}_1 * \mathbf{w} + \mathbf{v}_2 * \mathbf{w} \]
\[ \mathbf{v} * ( \mathbf{w}_1 + \mathbf{w}_2 ) = \mathbf{v} * \mathbf{w}_1 + \mathbf{v} * \mathbf{w}_2 \]
\[V \otimes W = Z / S\]
\[\mathbf{v} \otimes \mathbf{w} = \mathbf{v} * \mathbf{w} + S\]
\end{definition}


This is a rather formal algebraic construction, however to 

\[\mathbf{v} \otimes \mathbf{w} \in V \otimes W\]
\[\mathbf{v} \in V \]
\[\mathbf{w} \in W \]
\[ c(\mathbf{v} \otimes \mathbf{w}) = c\mathbf{v} \otimes \mathbf{w} = \mathbf{v} \otimes c\mathbf{w} \]
\[ (\mathbf{v}_1+\mathbf{v}_2 ) \otimes \mathbf{w} = \mathbf{v}_1 \otimes \mathbf{w} + \mathbf{v}_2 \otimes \mathbf{w} \]
\[ \mathbf{v} \otimes ( \mathbf{w}_1 + \mathbf{w}_2 ) = \mathbf{v} \otimes \mathbf{w}_1 + \mathbf{v} \otimes \mathbf{w}_2 \]


As it turns out, a basis of this product space is all the tensors between basis vectors of the two lienar spaces.

As it turns out $\mathbb{R}^n \otimes V \cong V^n$


\section{Grassmann algebra}

Tensor theory will become fruitful to the study of the Grassman algebra

The Grassman algebra algebraically capturing geometric information such as area, volume, and orientation formed by vectors.

\begin{definition}[Grassman algebra]
	A \emph{Grassman algebra over $(V,K,+,\cdot)$} or \emph{exterior algebra over $(V,K,+,\cdot)$} is the quotient ring $\bigwedge (V) = T(V) / I$
	\begin{itemize}
		\item $T(V)$ is the tensor algebra on $(V,K,+,\cdot)$
		\item $I$ is the ideal formed by $x \otimes x$ where $x \in V$
	\end{itemize}
\end{definition}

This definition relies heavily on ring theory, but what does it characterize?
It is all about introducing the wedge product; an operation that generalizes the cross product to arbitrary linear spaces.
The central rule encoded into the algebra is that $\mathbf{v} \times \mathbf{v} = \mathbf{0}$.
It is not a commutative algebraic structure, however we can prove that it is anticommutative.

Wedge product
k-blade

Given a Grassmann algebra $\bigwedge (V)$, the subspace of $k$-blades is represented as $\bogwedge_{k} (V)$

bivector
multivector
Hodge star operator

















\chapter{Tensors}


The demands of physics have required more complicated classes of linear functions. The mathematical attempt to classify them has been by the invent of \emph{tensors}.

We are at least familiar with scalars, vectors, linear maps, multilinear maps, linear forms, bilinear forms, and multilinear form. These are indeed all examples of different types of tensors.

\section{Vectors and covectors (linear forms)}
$[\mathbf{x}]_k$ means the scalar at row $k$ of the column vector $\mathbf{x}$
$[\mathbf{A}]_{ij}$ means the scalar at row $i$, column $j$ of the matrix $\mathbf{A}$

$T(\mathbf{x}) = \mathbf{A}\mathbf{x}$
$\mathbf{f}_{k} = \mathbf{e}_{i}$

Consider a linear space $V$ with an old basis $\mathbf{e}_i$ and new basis $\mathbf{f}_i$. We use some rule of the following form to translate each $\mathbf{e}_i$ to the new basis (allowing the transformation of any vector to the new basis).
\[\mathbf{f}_i = \sum_{j=1} ^{n} [\mathbf{P}]_{ij} \mathbf{e}_j \]
\[\mathbf{e}_i = \sum_{j=1} ^{n} [\mathbf{P}^{-1}]_{ij} \mathbf{f}_j \]

contravariant
\[\mathbf{P}^{-1}[\mathbf{v}]_{\mathcal{E}} = [\mathbf{v}]_{\mathcal{F}}\]
Vectors have the contravariant property

covariant
\[\mathbf{P}[\phi]_{\mathcal{E}} = [\phi]_{\mathcal{F}}\]
Linear forms (covectors) have the covariant property



\section{Ricci calculus}


Tensors are quite complex objects, so we introuce some notation that compactifys expressions while retaining all necessary information of the tensor.



\begin{definition}[Tensor (indexed array definition)]
A \emph{$(p,q)$ tensor} is a 
\end{definition}


\subsection{Indexing laws}
Superscript is used for contravariant coefficients 
Subscript is used for covariant coefficients 

\subsection{Einstein notation}

Einsteun notation is essentially the omission of the Sigma summation symbol when the context is clear.


\[\sum_{i} \mathbf{e}_i x^{i}\]

The main idea is that \emph{sums are implicitly implied over an index when you have a subscripted symbol times a superscripted symbol}.
These expressions are so common that it becomes well worth introducing this notation.

the summation index may not appear more than twice to invoke Einstein notation.



\section{Tensor}

We have investigated scalars and vectors, and we've had the epiphany that matrixes represent linear maps on vectors; however just as we create an algebra upon matrixes (linear maps), can we form an algebra on 'tensors' (multilinear maps)?

\begin{definition}[Tensor (multilinear map definition)]
Let $(V,F,+,\cdot)$ be a linear space. A \emph{$(p,q)$ tensor} is a multilinear map that takes $p$ covectors from $V^{*}$ and $q$ vectors from $V$ as arguments, and outputs a scalar.
\end{definition}

We know that a linear map can be defined pureply by knowing how it transforms a linear space's basis elements (hence why its matrix representation is possible).

The same holds for tensors, and it is why some think of tensors as a possibly multidimensional array (even if that explanation doesn't reveal a tensor's behaviour).

\begin{proposition}[Tensors as an array]
	Let $(V,F,+,\cdot)$ be a linear space with a $(p,q)$ tensor $T$. Define the array of scalars $T^{i_1 \hdots i_p}_{j_1 \hdots j_q} = T(\varepsilon^{i_1}, \hdots , \varepsilon^{i_p}, \mathbf{e}_{j_1} , \hdots , \mathbf{e}_{j_q})$.

\end{proposition}


%We have seen matrixes define linear maps on vectors, where an one dimension of the output vector may be linearly affected by all dimensins of the input vector. Hence the tensors define \emph{multilinear maps}, and can also define \emph{multilinear} equations just like how matrixes define linear equations.






\begin{definition}[Tensor product]
For two linear spaces $V,W$, the \emph{tensor product} is the linear space $V \otimes W$ associated with a bilinear map  $T : V \times W \to V \otimes W$
\end{definition}


We can create an equivalent but more abstract definition of a tensor in this way

\begin{definition}[Tensor (tensor product definition)]
A $(p,q)$ tensor is an element of a tensor product space $\bigotimes^{p}_{n=1} V^{*} \otimes \bigotimes^{q}_{n=1} V$
\end{definition}






\begin{definition}[Tensor power]
\end{definition}


\begin{definition}[Tensor algebra]
	\[\]
\end{definition}








\chapter{More eigenequations}

minimal polynomial