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UID:www.tcs.tifr.res.in/event/1441
DTSTAMP:20240722T043611Z
SUMMARY:Equivalence Testing of Principal Minors
DESCRIPTION:Speaker: Roshan Raj (IIT Bombay)\n\nAbstract: \nFor a square ma
trix A with n rows and n columns and a subset S of [n]\, the corresponding
principal minor is the determinant of the submatrix of A with rows and co
lumns indexed by elements in S i.e. det(A[S\, S]). Two operations on a mat
rix A that preserve all the principal minors are:\nTaking the transpose of
the matrix.\nMultiplying it by an invertible diagonal matrix on one side
and by its inverse on the other.\nThe matrices DAD^{-1} and DA^TD^{-1} are
called diagonally similar to A when D is an invertible diagonal matrix.\n
For a square matrix A of dimension n\, a subset S of [n] of size at least
two and at most n-2 is called a cut if the rank of both the submatrices A[
S\, [n]-S] and A[[n]-S\, S] is less than two. In a couple of works\, Rapha
el and Loewy showed that for an irreducible matrix A with no cuts\, anothe
r matrix B has the same principal minors if and only if B is diagonally si
milar to A. \nIn this work\, we give an extension of their result by givi
ng a complete characterization of when two matrices can have the same prin
cipal minors and give a polynomial time algorithm to test it. We also pres
ent some applications of equivalence testing of principal minors in Combin
atorics and Polynomial Identity Testing.\nThis is joint work with Abhranil
Chatterjee\, Rohit Gurjar\, and Sumanta Ghosh.\nShort Bio:\nRoshan Raj
is a PhD student in the CSE department at IIT Bombay working under the me
ntorship of Prof. Rohit Gurjar. He holds a Betch degree in CSE from IIT B
HU.\n
URL:https://www.tcs.tifr.res.in/web/events/1441
DTSTART;TZID=Asia/Kolkata:20240802T160000
DTEND;TZID=Asia/Kolkata:20240802T170000
LOCATION:via Zoom in A201
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