Jerzy Baksalary

Jerzy Kazimierz Baksalary (25 June 1944 – 8 March 2005) was a Polish mathematician who specialized in mathematical statistics and linear algebra. In 1990 he was appointed professor of mathematical sciences. He authored over 170 academic papers published and won one of the Ministry of National Education awards.

Biography
He was a graduate of the Faculty of Mathematics, Physics and Chemistry at the University of Adam Mickiewicz in Poznan (1969). In the years 1969-1988 he was associated with the Department of Mathematics of the University of Agriculture in Poznań.

From 1996, he was the dean of the Faculty of Mathematics, Physics and Technology at the Military University of WSP, and after the WSP joined the Zielona Góra University of Technology and the emergence of the University of Zielona Góra, he headed the Linear Algebra and Mathematical Statistics group.

Research
In 1979, Baksalary and Radosław Kala proved that the matrix equation $$AX - YB = C$$ has a solution for some matrices X and Y if and only if $$(I - A^-A)C(I - B^-B) = 0$$. (Here, $$A^-$$ denotes some g-inverse of the matrix A.) This is equivalent to a 1952 result by W. E. Roth on the same equation, which states that the equation has a solution iff the ranks of the block matrices $$\begin{bmatrix} A & C\\ 0 & B\\ \end{bmatrix}$$ and $$\begin{bmatrix} A & 0\\ 0 & B\\ \end{bmatrix}$$ are equal.

In 1980, he and Kala extended this result to the matrix equation $$AXB + CYD = E$$, proving that it can be solved if and only if $$K_GK_AE = 0, K_AER_D = 0, K_CER_B = 0, ER_BR_H = 0$$, where $$G := K_AC$$ and $$H := DR_B$$. (Here, the notation $$K_M := I - MM^-$$, $$R_M := I - M^-M$$ is adopted for any matrix M. )

In 1981, Baksalary and Kala proved that for a Gauss-Markov model $$\{y, X\beta, V\}$$, where the vector-valued variable has expectation $$X\beta$$ and variance V (a dispersion matrix), then for any function F, a best linear unbiased estimator of $$X\beta$$ which is a function of $$Fy$$ exists iff $$C(X)\subset C(TF')$$. The condition is equivalent to stating that $$r(X\vdots TF') = r(X)$$, where $$r(\cdot)$$ denotes the rank of the respective matrix.

In 1995, Baksalary and Sujit Kumar Mitra introduced the "left-star" and "right-star" partial orderings on the set of complex matrices, which are defined as follows. The matrix A is below the matrix B in the left-star ordering, written $$A ~*< B$$, iff $$A^*A = A^*B$$ and $$\mathcal{M}(A)\subseteq \mathcal{M}(B)$$, where $$\mathcal{M}(\cdot)$$ denotes the column span and $$A^*$$ denotes the conjugate transpose. Similarly, A is below B in the right-star ordering, written $$A <*~ B$$, iff $$AA^* = BA^*$$ and $$\mathcal{M}(A^*) \subseteq \mathcal{M}(B^*)$$.

In 2000, Jerzy Baksalary and Oskar Maria Baksalary characterized all situations when a linear combination $$P = c_1P_1 + c_2P_2$$ of two idempotent matrices can itself be idempotent. These include three previously known cases $$P = P_1 + P_2$$, $$P = P_1 - P_2$$, or $$P = P_2 - P_1$$, previously found by Rao and Mitra (1971); and one additional case where $$c_2 = 1 - c_1$$ and $$(P_1 - P_2)^2 = 0$$.