Let \(\mathbb{F}\) be an arbitrary ﬁeld. An element \(x\) of an \(\mathbb{F}\)-algebra is called quadratic when it is annihilated by a polynomial of degree \(2\) with entries in \(\mathbb{F}\). Such elements include the involutions (\(x^2=1\)), the idempotents (\(x^2=x\)), the square-zero elements (\(x^2=0\)), quarter turns (\(x^2=-1\)) and so on.

Starting from the 1960’s, decomposing matrices into quadratic ones has attracted the attention of many researchers, for decomposition into sums as well as decompositions into products. Most notably:

- products of idempotents have been studied by Erdos [3] and Ballantine [1];
- products of two involutions have been characterized by Wonenburger [8], Djoković [2], Hoﬀmann and Paige [5]; Gustafson et al [4] have proved that every matrix with determinant \(\pm 1\) is the product of at most \(4\) involutions, and no less in general;
- sums of idempotents have been characterized by Wu [10].

This talk will focus on recent breakthroughs in such problems. One of the main ones deals with the so-called “mixed length \(2\) problem”, for which a complete solution has recently been found [6]. In the mixed length \(2\) problem for sums (respectively, for products), one considers arbitrary ﬁxed polynomials \(p\) and \(q\) with degree \(2\) over \(\mathbb{F}\), and one asks which square matrices split into \(A+B\) (respectively, \(AB\)) for matrices \(A\) and \(B\) such that \(p(A)=0\) and \(q(B)=0\). Many results on the mixed length \(2\) problem were obtained by J.-H. Wang in the early 1990’s, but he stuck to considering matrices over the complex numbers [9], which hides most of the diﬃculties that arise in the general case.

We will also point to similar decomposition problems in diﬀerent contexts: stable decompositions (see e.g. [7]), decompositions of endomorphisms of inﬁnite-dimensional vector spaces, decompositions into sums of selfadjoint or skew-selfadjoint endomorphisms, decompositions in orthogonal or symplectic groups.

[1] C. S. Ballantine, Products of idempotent matrices, Linear Algebra Appl. 19:81–86, 1967.

[2] D. Ž. Djoković, Products of two involutions, Arch. Math. 18:582–584, 1967.

[3] J. Erdos, Products of idempotent matrices, Glasgow Math J. 8(2):118–122, 1967.

[4] W.H. Gustafson, P.R. Halmos, H. Radjavi, Products of involutions, Linear Algebra Appl. 13:157–162, 1976.

[5] F. Hoﬀman, E. C. Paige, Products of two involutions in the general linear group, Indiana Univ. Math. J. 10:1017–1020, 1971.

[6] C. de Seguins Pazzis, The sum and the product of two quadratic matrices, ArXiv, 2017.

[7] C. de Seguins Pazzis, Products of involutions in the stable general linear group, J. Algebra 530:235–202, 2019.

[8] M. J. Wonenburger, Transformations which are products of two involutions, J. Math. Mech. 16:327–338, 1966.

[9] J.-H. Wang, Sums and products of two quadratic matrices, Linear Algebra Appl. 129-1:127–149, 1995.

[10] P.Y. Wu, Sums of idempotent matrices, Linear Algebra Appl. 142;43–54, 1990.

Université de Versailles Saint-Quentin-en-Yvelines

Email address: dsp.prof@gmail.com