Jerzy Baksalary Explained

Jerzy Baksalary
Birth Date:	25 June 1944
Birth Place:	Poznań, Poland
Death Place:	Poznań, Poland
Discipline:	Mathematics
Workplaces:	University of Tampere University of Zielona Góra
Sub Discipline:	Linear algebra Mathematical statistics
Alma Mater:	Adam Mickiewicz University

Jerzy Kazimierz Baksalary (25 June 1944 – 8 March 2005) was a Polish mathematician who specialized in mathematical statistics and linear algebra.^[1] 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.^[2]

Biography

Early life and education (1944 – 1988)

Baksalary was born in Poznań, Poland on 25 June 1944. From 1969 to 1988, he worked at the Agricultural University of Poznań.

In 1975, Baksalary received a PhD degree from Adam Mickiewicz University in Poznań; his thesis on linear statistical models was supervised by Tadeusz Caliński.^[3] He received a Habilitation in 1984, also from Adam Mickiewicz University, where his Habilitationsschrift was also on linear statistical models.

Career (1988 – 2005)

In 1988, Baksalary joined the Tadeusz Kotarbiński Pedagogical University in Zielona Góra, Poland, being the university's rector from 1990 to 1996. In 1990, he became a "Professor of Mathematical Sciences", a title received from the President of Poland. For the 1989–1990 academic year, he moved to the University of Tampere in Finland. Later on, he joined the University of Zielona Góra.

2005 death and legacy

Baksalary died in Poznań on 8 March 2005. His funeral was held there on 15 March 2005. There, Caliński praised Baksalary for his "contributions to the Poznań school of mathematical statistics and biometry".

Memorial events in honor of Baksalary were also held at two conferences after his death:

• The 14th International Workshop on Matrices and Statistics, held at Massey University in New Zealand from 29 March to 1 April 2005.
• The Southern Ontario Matrices and Statistics Days, held at the University of Windsor^[4] in Canada from 9 to 10 June 2005.

Research

AX - YB = C has a solution for some matrices X and Y if and only if (I - A ^{-A)C(I} - B ^{-B)} = 0 .^[5] (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} .^[6] (Here, the notation K _{M} : = I - MM ^{-} , R _{M} : = I - M ^{-M} is adopted for any matrix M.)\{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( ⋅ ) denotes the rank of the respective matrix.^[7]

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 l{M}(A)\subseteq l{M}(B) , where l{M}(⋅) denotes the column span and A ^{*} denotes the conjugate transpose.^[8] Similarly, A is below B in the right-star ordering, written A <*~ B , iff AA ^{*} = BA ^{*} and l{M}(A ^{*)} \subseteq l{M}(B ^{*)} .P = c _{1P} _{1} + c _{2P} _{2} of two idempotent matrices can itself be idempotent.^[9] 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 .

Notes and References

1. Baksalary. Oskar Maria. Styan. George P. H.. 2005-11-15. Some comments on the life and publications of Jerzy K. Baksalary (1944–2005). Linear Algebra and Its Applications. Tenth Special Issue (Part 2) on Linear Algebra and Statistics. en. 410. 3–53. 10.1016/j.laa.2005.08.011. 0024-3795. free.
2. Web site: Biografia w „Głosie Uczelnianym Uniwersytetu Zielonogórskiego” . dead . https://web.archive.org/web/20120911155458/http://www.uz.zgora.pl/wydawnictwo/miesiecznik04-2005/04.pdf . 2012-09-11 . 2018-11-21 . www.uz.zgora.pl . pl.
3. Baksalary . Oskar Maria . Styan . George P.H. . 2005 . Jerzy K. Baksalary (1944–2005) and his contributions to Image . Image . en . 34 . 14–15.
4. Web site: 2005 . Southern Ontario Matrices and Statistics Days Program . homepages.tuni.fi.
5. Baksalary . J.K. . Kala . R. . June 1979 . The matrix equation AX − YB = C . Linear Algebra and its Applications . en . 25 . 41–43 . 10.1016/0024-3795(79)90004-1.
6. Baksalary . J.K. . Kala . R. . April 1980 . The matrix equation AXB+CYD=E . Linear Algebra and its Applications . en . 30 . 141–147 . 10.1016/0024-3795(80)90189-5.
7. Baksalary . J. K. . Kala . R. . July 1981 . Linear Transformations Preserving Best Linear Unbiased Estimators in a General Gauss-Markoff Model . The Annals of Statistics . 9 . 4 . 913–916 . 10.1214/aos/1176345533 . 0090-5364.
8. Baksalary . Jerzy K. . Mitra . Sujit Kumar . 1991-04-15 . Left-star and right-star partial orderings . Linear Algebra and its Applications . 149 . 73–89 . 10.1016/0024-3795(91)90326-R . 0024-3795.
9. Baksalary . Jerzy K. . Baksalary . Oskar Maria . December 2000 . Idempotency of linear combinations of two idempotent matrices . Linear Algebra and its Applications . 321 . 1-3 . 3–7 . 10.1016/s0024-3795(00)00225-1 . 0024-3795.