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\begin{quote}  In 1888, Italian mathematician Giuseppe Peano formalized the first modern definition of  vector spaces and linear transformations. Peano's formal definition of vector spaces did not gain  much attention or popularity until 1918 when Weyl "essentially repeated Peano's axioms" in his book Space-Time-Matter, and articulated an important relationship between a “systems view”  and a “vector spaces and transformations view” of linear algebra. "Weyl then brings the subject  of linear algebra full circle, pointing out that by considering the coefficients of the unknowns in a 
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\paragraph{Cauchy (1829}  Possibly the first paper to couch eigenvalue problems in terms of matrices (as  opposed to differential equations and eigenfunctions). Symmetric matrices have real eigenvalues. 
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A very curious thing: "outer or $\times$ m[ultiplication] of spaces, vectors, operators" shows up in the index of the English translation~\cite{Weyl1931}, but the word ``outer'' is absent on the cross-referenced pages both in the English and German editions. The original German editions don't have an entry with ``outer'' at all. So we must conclude that this was an invention of the English translator, who is himself a notable American physicist.  \paragraph{Bocher (1931)}  Bocher 1931 appears to have the first inkling that matrices are themselves quantities in their own right rather than simply a collection of elementary quantities - section 21 on "Matrices as complex quantities"  \paragraph{Turnbull and Aitken (1932)}  Emphasis on systematic notation for quadratic and bilinear forms as $x'Ay$. 
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citation being Zehfuss, 1858. [3] refers to Gibbs's terminology of  dyad, also indeterminate product.  \paragraph{Bartlett (1934)}  For the  inner or scalar product $\sum x_1 x_2 $  one common notation is $(S_1 S_2)$; but  since we may regard $S_1$ as a matrix with one row, and $S_2$ the  transposed matrix of $S_2$ with one column, it will be convenient here  to write this product $S_1 S_2^\prime$, the usual matrix multiplication being  understood.  \url{http://dx.doi.org/10.1017/S0305004100012512}  \paragraph{Turnbull (1934)}  Turnbull 1934 appears to have popularized the notion of row/column vectors/matrices - he references Turnbull and Aitken (requested)  \begin{quote}  p14: \url{https://archive.org/stream/vectorsofmindmul010122mbp#page/n35/mode/2up}  "A matrix that consists of a single column will be called a \textit{column vector}. A  matrix that consists of a single row will be called a \textit{row vector}. The  vectorial terminology is probably due to the fact that the elements in any  array may be regarded as the Cartesian coordinates of a point in a space of as  many dimensions as there are elements in the array. This point, together with  the original, determines a direction in space. In this manner any array of a  matrix can be given a vectorial interpretation."  \end{quote}  Turnbull also uses capitals for matrices and lowers for column vectors.  The idea of row*column = scalar product goes all the way back to Cayley! although he did not use these names and he spelled out the elements.  doi: 10.1098/rstl.1858.0002  \paragraph{Bartlett (1938)}  Notation to simplify writing col vector, write them as row vector transposed  "In conformity with orthodox notation, vectors may sometimes for clarity be denoted by small letters, x, being a column vector, x' a row vector" - Bartlett, 1938 \url{https://1988c0fbd715ddc21e4eab393a153f2278df184c.googledrive.com/host/0B8_joYJa2eNzfkNIQ2ZiY1hxS0FlMTI4ZC1IYnRUZ0ttd0NrQlYxcEdMZnFSMnpYZ3ltbUE/v34/1/S0305004100019897.pdf}  \url{http://dx.doi.org/10.1017/S0305004100019897}  Bartlett 1934 (referenced by 1938) did not use this notation, but did use the notation $x^\prime y$ for inner product  and also was among the first to use the notion of vector transpose.  Other people who ahad the idea of a vctor transpose:  Thomson, 1936  \url{http://psycnet.apa.org/journals/edu/27/1/37/}  \paragraph{Schwerdtfeger (1938)~\cite{Schwerdtfeger1938}}  A book about matrix functions. 
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\paragraph{Ingram (1944)}  Outer products  TODO \url{http://www.jstor.org/stable/3029991}  TODO \url{http://www.jstor.org/stable/2307548}  In  \url{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078730/}  they explain their notation is a variant of Birkhoff and Langer, 1923  Row vector $\cdot v$, column vector $u:$  Inner product is $\cdot v u:$  Outer product is $u:\cdot v$  \paragraph{Schwerdtfeger (1945)}  \url{http://dx.doi.org.libproxy.mit.edu/10.1063/1.1707507}  A vector in the three-dimensional space will  be represented by a column matrix  where $xy'$ is the column-row product (representing  the often so-called "dyadic product" of the  two vectors $x$, $y$),  references his own Les Fonctions de Matrices I, Actualites  scientifiques (1938) - REQUESTED  TODO \url{http://www.jstor.org/stable/2303007} (1944)  TODO Householder 1950 \url{http://www.jstor.org/stable/2308297}  \paragraph{Wade (1948)~\cite{Wade1948}}  Cited in second edition of \cite{Margenau1943}. 
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\end{quote}  \paragraph{Milner (1952)}  \url{http://rspa.royalsocietypublishing.org/content/214/1118/312}  is apparently the first to express the "matrix product" of two vectors, expressed as $a \overline{a}$.  The terms scalar product $\overline{a} b$ and matrix product are introduced.  \paragraph{Householder (1953)~\cite{Householder1953}}  The first book on linear algebra focusing on numerical algorithms. 
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\end{quote}  \end{enumerate}  \subsection{OTHER IDENTITIES?}  Another identity $(Au - v)^T(Au - v) = u'Au -...$