History of representing linear transformations with matrices

Question

I'm working from the perspective of computer graphics and I'd like to understand how we came to use matrices to express transformations in $R^2$ and $R^3$ that represent things like rotation, scale, and skew and the composition of those transformations.

My understanding is that matrices first appeared as mere abbreviations for systems of linear equations. And that later Gauss showed that they could represent linear transformations as well. So I can see that at that point we could define matrices to represent arbitrary transformations. But it's not clear why this was important. Was it simply the convenience and compact representation?

And second, were matrix multiplication defined in such a way as to allow us to compose these transformations or was it defined already and it just happened that that allowed the transforms to be composed? That seems like magic.

If the matrix operations came later, then I also wonder what addition was intended to represent? I don't know of any useful geometric meaning.

Answer 1

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The short answer seems to be that the composition idea led to the method of multiplication. But at that point, Gauss wasn't thinking in terms of a "matrix algebra".

The term 'determinant' was first introduced by Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms. He used the term because the determinant determines the properties of the quadratic form. However the concept is not the same as that of our determinant. In the same work Gauss lays out the coefficients of his quadratic forms in rectangular arrays. He describes matrix multiplication (which he thinks of as composition so he has not yet reached the concept of matrix algebra) and the inverse of a matrix in the particular context of the arrays of coefficients of quadratic forms.

Later, Cayley formalized the notion of a matrix algebra in the abstract.

Cayley in 1858 published Memoir on the theory of matrices which is remarkable for containing the first abstract definition of a matrix. He shows that the coefficient arrays studied earlier for quadratic forms and for linear transformations are special cases of his general concept. Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses. He gave an explicit construction of the inverse of a matrix in terms of the determinant of the matrix. Cayley also proved that, in the case of 2 × 2 matrices, that a matrix satisfies its own characteristic equation. He stated that he had checked the result for 3 × 3 matrices, indicating its proof, but says:- I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree. That a matrix satisfies its own characteristic equation is called the Cayley-Hamilton theorem so its reasonable to ask what it has to do with Hamilton. In fact he also proved a special case of the theorem, the 4 × 4 case, in the course of his investigations into quaternions.