Lecture39 Determinants minors permanents 2390

Red Given 2actor spaces V W on a field1K a linear map VFWwe have 3 natural linear maps for all u o

i P v Tif fi I W with Fln un fan far

e S v Is S W with S fn vn tiny fun

A v Mf n W with A G in n ang fun aGun

1 is defined via universal property 2 G an obtained frm FIG

by obning f I am un v uit nix uphis in

2 1 h it

ri Un E VCw Wh I w wit wi on it new

f1I.mwontp.fim

demensinafik vector spaces Liv Weimar

Say V elk We 1km pick bases In V W so

GE Matman k

m NTV f AKIN12 6 12 basis

1111k 1K'dis j C jus nG E Mataya k is tic einem

Def Mines of f are the entries of thematrix for gMoreprecisely write I in sin E X k subsetsMcm

I ji Ou G E C ChN Li NY

139

N fI j

det FI DI fsub matrix offwith rows incolumns in jHowdo we compute Nlt

Fix B 3 vi o un f a basis forVB 3wi own E W

Write fluj Ei ai j wiAF vg n nuju

14 aij wild a É aijuwiN G I I weft of wi n n win in A f lyin anju

Recall Wren n awrong sign o wi n nwin t resin

So cuffofwi n win in 1 1 aij wild h É aijuwi A

is I sign aim j hire ji hiringJuTESL

In particular If ken m write ditch DIII sThis recovers the permutation formula for determinants

ditA s Eg Simo are 9th z 9 ruin

E

I SnMl 6 918in 926cg a nan

Consequence dit A dit AT AG Matan IK

13930Conscience Row expansion formula for dit A

PH Fix ith Row MA

at A Éaijfgpsmldarwgj.hr 9nn

restrict tho 5 31 J nty

si it and so

sign o a ti sign FSo A def Alois g its

out A tnlitjaij.atA D

Qbs Using we get column expansion formula as well

Consequent If Ahas z equal rows cuss cereal whythen dit A 0

PH N G vi n nvm O in A IN IK by K D

Def Cofactormatrix of ACot A 1 J at A Lei jen

Conference CfAJ A ditch In Alot AT

PH LGATA It ColbieoneI unite itAll aej

13990If i j this is jth column expansion of detA

If it att A where

A is the matrix obtainedfrom A by replacing nth of Abythe

jth all of A By Consequence dit A o

so Cf ATA detA In

A lotAT WA AT GfADT A I CatAt IntdetA In

HofAt cy't's at At G ti at A of A g iA

Legume determinant is multiplicative seeHWI2

Roof Write V W W U dunt dunW dunlin

B 3u unE Be 3W i Wnt B 3 u unE basesInV War

Then A got A V N W N'N

Sp tin nun Splwin nun Splain aunRIK Atl I the I

Qdit got

map lls 1kso dit got dit s dit f gives the linearon the bottom row

52Permanents

Again fix U W with dunlin din W m V f W linear

Q What happens if we do this to 5 v 54s W

I We get permanents HW12 13950

Def Perm f I j coffof Wi win inStacy ijnd c c ik dis s ju

We are notallowed to have repetitions so we are notcapturing

all the coefficients of 5 f Can includethisby repeatingcolumns

In particular for n m k we have

Pum A Jfs a.ru ii a

renin

D It is no longer true that matrices with repeated rowshave permanent so This makes it say hardto amplepermanents

In particular there are no good algorithms for computing permanents

Best results are dueto A Barrinok

ÉÉgkE1ft Wesay X admits a GaussicDempositin if

X X XoXt

where X diagonal matrix

x 18 5ai

II Gaussiandecompositions are unique

PH X X Xt Y yo y t

7 X X Xt Y'T Y Gaussiandump of Y

since 7 X 18 o t o Ii l'dxt 174 I's I's s ai 29 it

So it'senough to show it fr diagonalmatrices

Gin X X Xt Y diagonal XIXIInPHILY Ifl

so both are diagonalinvertible

x in xoxÉ 1 yy

yn forces X Y X 2 9,2 Xu yXt l 912 9in

y sayino x'xtij

Elyon4th

j i x iiXDijXo Mt ij

forces A ijo f j i so XI In

Similarly X In so X 7

Thurs X admit G D D x to fit in

THE A'IÉx D X X Xt

a lift Hit Fitt

It to ai di da

so dit X di du to c all di to

D II A di du nonzero th

Moura d Di xdid D n D dz D

yIn gural du 7ÉÉÉÉSo Xo is uniquely determined by theprincipal minors ofX

e Explicitly write e anD X ut

IIIIIX Jj s g

b i i j

ask.IExDji D

i

iand duck X X X Xt by induction mu

Qbs The construction is also true for Glu R where R is not

necessarily a imitative ring I define dits via column expansionObs Nmvanishingminors is an openaudition in MatuallkWe have a denseopen set U E Matu Ilk whichwe parameterizeas

BigBruhat UE IK E x Aall

entries mx entriesmyo entriesmXtCanprove statements on Matan us by restricting to U