Basic Maths. Fiorella Sgallari University of Bologna, Italy Faculty of Engineering Department of Mathematics - CIRAM

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1 Bsic Mths Fiorell Sgllri Uiversity of Bolog, Itly Fculty of Egieerig Deprtmet of Mthemtics - CIRM

2 Mtrices Specil mtrices Lier mps Trce Determits Rk Rge Null spce Sclr products Norms Mtri orms Positive defiite mtrices M-mtrices Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog

3 Mtrices Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 3

4 Mtrices Squre Mtri m = submtri Defiitio Let be mtri m Let i < i < < i k m d j < j < < j l two sets of cotiguous idees The mtri S(k l) of etries s pq = ipjq with p =,, k, q =,, l is clled submtri of If k = l d ir = jr for r =,, k, S is clled pricipl submtri of Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 4

5 Mtrices Symmetric mtrices = T tisymmetric mtrices = - T Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 5

6 Mtrices Mtri sum Mtri multiplictio by sclr Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 6

7 Mtrices Mtri product: Give d where The squre mtrices for which the property B = B holds, will be clled commuttive Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 7

8 Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 8 Specil Mtrices u u u u u u u u u u U Upper trigulr l l l l l l l l l L Lower trigulr

9 Specil Mtrices H H h h h h h h h h h h h h h h h h h h3 h3 h3 h33 h h h h Upper Hesseberg Lower Hesseberg Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 9

10 Bded mtrices bdwidth p+q+ p=q= tridigol mtri p=q= petdigol mtri p right hlf-bdwidth q: ij = per j>i+q Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog q left hlf-bdwidth p: ij = per i>j+q

11 Digolly domit Mtrices mtri R is clled digolly domit by rows if for i=,,, d t lest for oe ide i ii ij j ji ii ij j ji is clled strictly digolly domit if ii ij j ji We hve lso digolly domit by colums Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog

12 Permuttio mtrices Differ from Idetity mtri by chge of rows P P P row echges colum echges Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog

13 Mtrices I Idetity D d d d33 d Digol Defiitio squre mtri of order is clled ivertible (or regulr or osigulr) if there eists squre mtri B of order such tht B =B = I B is clled the iverse mtri of d is deoted by mtri which is ot ivertible is clled sigulr If is ivertible its iverse is lso ivertible, with ( ) = Moreover, if d B re two ivertible mtrices of order, their product B is lso ivertible, with ( B) = B Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 3

14 Mtrices ( T ) T =, (+B) T = T +B T, (B) T = B T T d () T = T R If is ivertible, the lso ( T ) = ( ) T = T Defiitio Let C m ; the mtri B = H C m is clled the cojugte trspose (or djoit) of if b ij = ā ji, where ā ji is the comple cojugte of ji Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 4

15 Mtrices mtri is clled orthogol if T = T = I, tht is = T Defiitio mtri C is clled hermiti or self-djoit if T = Ā, tht is, if H =, while it is clled uitry if H = H = I Filly, if H = H, is clled orml s cosequece, uitry mtri is oe such tht = H Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 5

16 Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 6 Sclr Product or sclr product o vector spce V defied over K ( ) is y mp <, > ctig from V V ito K which ejoys the followig properties: C,,, C y y y,,, 3 C C z y z y z z y,,,,,, Emples i i i T i y y y, i i i H C i y y y,

17 Sclr product Moreover, for y give squre mtri of order d for y, y C the followig reltio holds (, y) = (, H y) I prticulr, sice for y mtri Q C, (Q,Qy) = (,Q H Qy), oe gets Property Uitry mtrices preserve the Euclide sclr product, tht is, (Q,Qy) = (, y) for y uitry mtri Q d for y pir of vectors d y Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 7

18 Norms Let V be vector spce over K We sy tht the mp from V ito R + U {} is orm o V if the followig ioms re stisfied:, C C C 3 y y, y C Emples i i i -orm m i,,, i Ifiity Norm - Chebyshev, T i i i -orm - euclide Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 8

19 Norms -Emples -orm i i i m i,,, i Ifiity Norm - Chebyshev, T i i i -orm - euclide = Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 9

20 Norms y vector orm defied o V is cotiuous fuctio of its rgumet Defiitio Two orms two positive costts such tht C ', '' re equivlet if there eist,, '' ' '' Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog

21 mtri orm is mppig such tht: Mtri orm : C, C C C 3 B B, BC 4 lso B B, B C No essezile m : C le orme uste soddisfo quest proprietà Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog

22 Mtri orm Defiitio We sy tht mtri orm is comptible or cosistet with vector orm if, R Defiitio We sy tht mtri orm is sub-multiplictive if R m, B R m q B B Theorem Let be vector orm sup The fuctio is mtri orm clled iduced mtri orm or turl mtri orm m Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog

23 i orm m i Mtri orm-emples j,,, i ij m i,,, i Ifiity Norm - Chebyshev m i,,, j ij i i -orm -Euclide T ( )?? Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 3 () spectrl rdius of mimum module of the eigevlues of

24 Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 4 j i ij F, Frobeius or Schur orm Mtri orms re equivlet ij j i ij j i,, m m F F Mtri orm

25 Mtri orm If is turl mtri orm, the I If is simmetric mtri, the T ( ) ( ) ( ) ( ) m If the mtri is simmetric d positive defiite, the m Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 5

26 Positive defiite mtrices Defiitio mtri C is positive defiite i C if the umber <, > is rel d positive C, mtri R is positive defiite i R if <, > > R, If the strict iequlity is substituted by the wek oe ( ) the mtri is clled positive semidefiite Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 6

27 Positive defiite mtrices Defiitio Let R The mtrices S =½( + T ), SS =½ ( T ) re respectively clled the symmetric prt d the skew-symmetric prt of Obviously, = S + SS If C, the defiitios modify s follows: S = ½( + H ) d SS =½ ( H ) Property rel mtri of order is positive defiite iff its symmetric prt S is positive defiite Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 7

28 Positive defiite mtrices Property Let C (respectively, R ); if <, > is rel vlued C, the is hermiti (respectively, symmetric) immedite cosequece of the bove results is tht mtrices tht re positive defiite i C do stisfy the followig chrcterizig property Property squre mtri of order is positive defiite i C iff it is hermiti d hs positive eigevlues Thus, positive defiite mtri is osigulr Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 8

29 Positive defiite mtrices Property Let R be symmetric The, is positive defiite iff oe of the followig properties is stisfied: <, > > with R ; the eigevlues of the pricipl submtrices of re ll positive; 3 the domit pricipl miors of re ll positive (Sylvester criterio); 4 there eists osigulr mtri H such tht = H T H Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 9

30 M- mtrices Defiitio osigulr mtri R is M- mtri if ij for i j d if ll the etries of its iverse re oegtive M-mtrices ejoy the so-clled discrete mimum priciple, tht is, if is M-mtri d, the (where the iequlities re met compoetwise) Property mtri R tht is strictly digolly domit by rows d whose etries stisfy the reltios ij for i j d ii >, is M-mtri Numericl Methods Civil Egieerig F Sgllri Dept Mthemtics-CIRM, Uiversityof Bolog 3