# Origins behind different terms for the same thing in Linear Algebra

It seems that there are many terms in linear algebra that have multiple names. For example, unitary and orthogonal both refer to the same general idea, a Hermitian is essentially a self-adjoint matrix, invertible and nonsingular, and there are definitely more that I can't think of off the top of my head. I've noticed that I've seen terms like orthogonal and self-adjoint in more classes/texts that feel like they consider linear algebra from a more abstract algebraic perspective, while I've seen terms like unitary and Hermitian in physics and more applied settings.

I was wondering if there was some kind of history behind these terms? Why do we have multiple terms for the same thing? Is it just a coincidence that I've seen Hermitian and unitary in these kinds of settings, or was the subject simply considered with different motivations by different people studying different things? If this is the case, I would love it if anyone could suggest references for where these terms originated, and also if there are multiple motivations/perspectives for linear algebra, are there any references for the origins of these different motivations/perspectives? I recall one of my physics professors mentioning briefly that mathematicians and physicists had independently developed the same theory only to realize later that they had been working on the same thing all along. Is there anywhere I could learn more about that history? And are there fields other than linear algebra where this has happened?

• “Nonsingular” and “invertible” are not the same. A linear endomorphism $f$ is called nonsingular if $f(x)=0$ has only the trivial solution (i.e., if $f$ is injective); it is called invertible if it has an inverse (i.e., if it is bijective). The two properties are equivalent on a finite-dimensional vector space, but inequivalent otherwise. For linear maps or matrices that are nonsingular but not invertible, consider the right-shift map on the vector space of all infinite sequences of real numbers, or the matrix $\pmatrix{1&-1\\ 1&1}$ over the commutative ring $\mathbb Z/4\mathbb Z$. Jul 22, 2023 at 3:51

Hermitian

A Hermitian matrix is called Hermitian... simply to honour Charles Hermite, who first studied Hermitian forms in Comptes Rendus de l'Académie des Sciences, Paris, vol. 41 (1855), pp. 181-183.

Unitary

Unitary matrices were first introduced by Léon Autonne in the paper Sur l’hermitien. See Comptes Rendus des Séances de l’Académie des Sciences, Paris, vol. 133 (1901), pp. 209-213, or Rendiconti del Circolo Matematico di Palermo (1884-1940), vol. 16, pp. 104–128 (1902). Here are the relevant paragraphs:

Soit H une forme d’Hermite ($$j,k=1,2,\ldots,n$$), $$H(x,y)=\sum_{jk}h_{jk}y_jx_k,\quad h_{jk}=\overline{h}_{kj},\quad h_{jj}=\text{réel},$$ telle, par suite, que $$\overline{H'}=H$$. Si l’expression $$H(x,\overline{x})$$, toujours réelle, reste aussi constamment positive pur ne s’évanouir qu’avec tous les $$x$$ à la fois, M. Lœwy (Math. Ann., t. L, p.560) dit que $$H$$ est une forme définie d’Hermite; je dirai plus brièvement que $$\overline{H}$$ est un hermitien. Un hermitien fournit la substitution $$n$$ — aire hermitienne. $$H=\left|\begin{matrix}x_j&\dfrac{\partial H(x,y)}{\partial y_j}\end{matrix}\right|$$

Je me propose de donner les propriétés principales de l’hermitien et de l’hermitienne.

Nommons unitaire toute substitution $$n$$ — aire $$U$$ telle que $$\overline{U'}U=E$$, $$\overline{U'}=U^{-1}$$. Si l’on effectue le changement de variables marqué par une $$n$$ — aire $$R$$, l’hermitien $$H$$ devient $$\overline{R'}HR$$, tandis que l’hermitienne devient $$R^{-1}HR$$. Le nouvel hermitien ne fournit plus la nouvelle hermitienne, à moins que $$R$$ ne soit unitaire, ce que l’on supposera toujours.