In this document, if A:

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1 m I this docmet, if A: is a m matrix, ref(a) is a row-eqivalet matrix i row-echelo form sig Gassia elimiatio with partial pivotig as described i class.

2 Ier prodct ad orthogoality What is the largest possible magitde of 1,? By Cachy-Byakovsky-Schwarz ieqality, 1, 1. If 1 ad are orthogoal, fid. 1 By the properties of the -orm ad orthogoal vectors, , 1,,, 0,, 1 so. 1 1 Defie a agle betwee two vectors 1 ad. If, cos, the 1 1 cos, 1 1, so fid the iverse cosie of the give ratio. Two vectors 3 are colliear if 0 0 or 180 ad two vectors are orthogoal if 90 or 70. A collectio of vectors, 1, are ot mtally orthogoal ad ot ormalized. Apply the Gram-Schmidt algorithm. See the text book, bt i essece: 1. or i from 1 to do a. Set vi i ; b. Sbtract off the projectio of v i oto v ˆ j for each of the previos i 1 ormalized vectors: or j from 1 to i 1 do, v v vˆ, v v ˆ i i j i j c. Normalize the i th vector assmig that vi 0: i Set vˆ i v v i. If the vectors, 1, are liearly idepedet, this will prodce orthoormal vectors.

3 Liear idepedece Give a collectio of vectors v,..., 1 v i m, determie if they are liearly idepedet. Create the matrix V v1 v ad fid the row-eqivalet matrix ref(a) i row-echelo form. If the rak of V eqals, the vectors are liearly idepedet; otherwise, they are liearly depedet. Corollary If > m, the vectors mst be liearly depedet, for the maximm rak of V is m. The spa of a set of vectors v,..., 1 v V icldes all liear combiatios of these vectors, so all vectors of the form v v 1 1 where,, 1. Is this a sbspace of V? Yes. The sm of two liear combiatios of vectors mst still be a liear combiatio of these vectors, ad mltiplyig a liear combiatio of a set of vectors is still a liear combiatio of these vectors. Give a collectio of vectors v,..., 1 v i m, fid the dimesio of ad a basis for the spa. Create the matrix V v1 v ad fid the row-eqivalet matrix ref(a) i row-echelo form. The dimesio of the spa is the rak of A. A basis for the spa are those colms of A that correspod to colms i ref(a) that have leadig o-zero etries. Whe is a set of vectors a basis for their spa? A set of vectors forms a basis their spa if ad oly if the vectors are liearly idepedet. Give a set of liearly idepedet vectors,, 1, sppose we apply Gram-Schmidt. Is the spa of the orthoormal set idetical to the spa of the origial set? Yes.

4 Liear operators U V A A A, what properties mst A have for A to be described as liear? 1 1 for all vectors 1, U ad A A for all vectors U. That is, if a vector-space operatio is performed first i U after which A is applied is the same as if A is applied ad the vector-space operatio is the performed i V. U V is a liear mappig, which is the domai, the co-domai ad the rage of A? The domai of A is U ad the co-domai of A is V. The rage of A is the collectio of all images A of vectors U. U V is a liear mappig, is the rage a sbspace of V? If v1, v V that are i the rage of A, there mst exist 1, U sch that A1 v 1 ad A v. By the liear properties of A, A A A 1 1 v v, ad becase U is a vector space, 1 U, ths, A v v is also i the rage of A. Ths, the rage is a sbspace of V. U V is a liear mappig, what is the image of 0 U? As 0 U = 0 for ay vector i U, A0 A0 0 A 0, so A How do we fid the matrices associated with each of the row operatios? Apply the row operatio i qestio to the idetity matrix: U V 0 0. U V Row operatio Addig a mltiple of oe row oto aother Swap two rows Mltiplyig a row by a o-zero scalar Physical iterpretatio shear reflectio scalig Descriptio Represetatio Effect Iverse Addig times Row i oto Row j ;i j Swappig Rows i ad j Ri j Mltiplyig Row i by R rji, R ;i j r r i, i j, j r r i, j j, i 0 1 Ri j R ;i rii, 1 ;i R

5 Nll space ad rage of fiite-dimesioal liear mappigs m, fid the dimesio of ad a basis for the rage. Give A, fid the row-eqivalet matrix ref(a) i row-echelo form. The dimesio of the rage is the rak of A. A basis for the rage are those colms of A that correspod to colms i ref(a) that have leadig o-zero etries. m, fid the dimesio ad a basis for the ll space. Give A, fid the row-eqivalet matrix ref(a) i row-echelo form. The mber of free variables eqals the dimesio of the ll space, ad to fid a basis for the ll space, solve A 0, which is eqivalet to solvig ref m A 0, which ca be solved sig backward sbstittio. Give A: U V, if A v ad A 0 V 0, the A By the properties of liearity, v. 0 A A A A A v 0 v 0 v. m V V m Give A:, arge that the dimesio of the ll space pls the dimesio of the rage always eqals. Give A, fid the row-eqivalet matrix ref(a) that is i echelo form. Every colm i ref(a) that has a leadig o-zero etry adds oe dimesio to the rage, ad every colm i ref(a) that does ot have a leadig o-zero etry adds aother free variable, ad ths adds oe dimesio to the ll space.

6 Oe-to-oe ad oto Whe is a liear mappig oe-to-oe? Whe every vector i the rage has a iqe pre-image. Whe is a liear mappig oto? Whe every vector i the co-domai has at least oe pre-image. Whe is a liear mappig oe-to-oe ad oto? Whe every vector i the co-domai has a iqe pre-image. m, what are tests for either oe-to-oe oto oto? If ref(a) has o free variables, A is oe-to-oe. If there is oe or more free variables, A is ot oe-to-oe, it is mayto-oe. If rak(a) = m, the mappig is oto. If rak(a) < m, the image of U is a sbspace of V ot eqal to V. m, are there cases whe A is ot oe-to-oe or oto? If > m, A ca ever be oe-to-oe, bt it may be oto if rak(a) = m. If = m, A is either oe-to-oe ad oto, or either. It caot be oe bt ot the other. If < m, A ca ever be oto, bt it may be oe-to-oe if rak(a) =.

7 Matrices What are the diagoal etries of a m matrix? The diagoal etries of a matrix A are all etries a i,i where i = 1,, mi{m, }. Whe is a matrix pper triaglar? Whe is it lower triaglar? Whe is it diagoal? A matrix is pper triaglar is all etries below the diagoal are zero. A matrix is lower triaglar if all etries to the right of the diagoal are zero. A matrix is diagoal if all the etries off of the diagoal are zero. Diagoal matrices are the oly matrices that are simltaeosly both lower ad pper triaglar. is a permtatio matrix, describe its properties. A matrix is a permtatio matrix if ad oly if every row has exactly oe 1 ad each colm has exactly oe 1 ad all other etries are 0. is a permtatio matrix, what is the reslt of A for? If a i,j = 1, this moves the j th etry of to the i th etry. is a permtatio matrix, what is the iverse. The iverse of a permtatio matrix is its traspose.

8 vector m v. m ad yo have the PLU decompositio of A with A = PLU, how do yo solve A = v for a give target A = PLU, so we are solvig PLU = v. Mltiply both sides by the iverse (traspose) of P to get P T PLU = Id m LU = LU = P T v. Now, (LU) = L(U), so this is eqivalet to solvig L(U) = P T v. As is kow, so is U, so let s represet the kow U by y; that is, y = U. Ths, we have the system of liear eqatios represeted by Ly = P T v. The agmeted matrix of this system of liear eqatios is L T P v, ad as L is lower triaglar, we may se forward sbstittio to solve for y. Now that we have y, we ow are solvig the system of liear eqatios represeted by y = U, so the agmeted matrix of this system of liear eqatios is U y. As U is pper triaglar, we may se backward sbstittio to fid.

9 The determiat, the trace ad the iverse Give AB, :, arge that det(ba) = det(b) det(a). Give a regio R with a fiite ad o-zero volme vol(r), AR is the regio comprised of the image of each vector i R, ad by defiitio, vol(a(r)) = det(a) vol(r). Next, if B AR is the regio comprised of the images of each vector i AR, vol(b(a(r))) = det(b) vol(a(r)) = det(b) det(a) vol(r). Bt B AR BAR, ad therefore det(ba) = det(b) det(a). Give A: where A is either pper triaglar, lower triaglar or diagoal, fid the determiat of A. Mltiply the diagoal etries of A., fid the determiat of A. If = or 3, we may se the short-cts we leared i class. Otherwise, for > 3, give A, fid the PLUdecompositio of A. Record the mber P of row swaps that were reqired to prodce P ad mltiply the determiat of U by 1 P. Give A: tr A., fid the trace of A deoted The trace of A is the sm of the diagoal etries of A., approximate det Id A. or sfficietly small, det Id A 1 tr A.

10 id ad approximate the determiats of A ad B det A ; ad , so det A det B 1.13 ; ad B , so det B Show that Id Id 1. IdId Id, so Id Id 1. Show that if A is ivertible, the 1 1 A A AA Id By the properties of operator compositio ad iverses, A A A A A AA, ad therefore AA Id 1. BA A B. Show that if A ad B are ivertible, the A B BA A B B A A IdA A A Id Usig the properties of the iverse ad matrix compositio, ad therefore 1 BA A 1 B 1. 1 Show that if A are ivertible, the A 1 A A 1 A AA Id Id, ad therefore A A 1.,

11 Adjoits of liear mappigs Recall that the adjoit of a liear mappig A: U V is that mappig A : V U sch that A, v, A v for all U ad all v V. or A: R is the cojgate traspose. m R, the adjoit is the traspose, deoted T A. or A: C m C, the adjoit U V, show that A By the properties of the adjoit, A. A A A, v, v, v, ad therefore A A If A1, A : U V., show that A A A A. 1 1 By the properties of the adjoit ad compositio of liear mappigs,, A A v A A, v A A, v A, v A, v , A v, A v, A v A v, A A v Ths U A A A A. 1 1 V, show that A A. By the properties of the adjoit ad compositio of liear mappigs, A A A A A A A A, v, v, v, v, v, v, v. A. Ths

12 U V ad B : V W, show that BA AB. By the properties of the adjoit ad compositio of liear mappigs, Ths BA U BA BA BA A B A B A B, w, w, w, w, w, w. A B. 1 1 U ad A is ivertible, show that A A. By the properties of the adjoit ad compositio of liear mappigs,, Id, AA, A A, A, A, A A, bt as this is tre for all 1 ad, ths 1 A A Id 1, so A A 1. or A: U V, a vector v is orthogoal to all vectors i the rage of A if ad oly if v is i the ll space of A. v is orthogoal to all vectors i the rage of A A, v for all U, A v for all U v is i the ll space of A id the liear combiatio of vectors,..., m that best approximates a vector m 1 v. If U 1 is sch that rak U rak U based o whether ref U has zero or more tha zero free variables, respectively. v, there is either a iqe soltio or ifiitely may soltios Otherwise, if rak U rak U v, we mst fid the least-sqares soltio by solvig U U U v. There is either a iqe soltio or ifiitely may soltios based o whether ref UU has zero or more tha zero free variables, respectively.

13 Self adjoit ad skew adjoit liear operators A liear operator A: U U is self-adjoit if A we say that A is cojgate symmetric. A liear operator A: U U is skew-adjoit if A: C C, we say that A is cojgate skew-symmetric. Give a liear operator A: U U, show that By the properties of the adjoit ad operator additio, A. R R, we say that A is symmetric. C C, A A A. R A is self-adjoit. R, we say that A is skew-symmetric. If A A A A A A A A A A,,,,,,, ad therefore, A, A, A A, A A A A A A , so it is self-adjoit. Give a liear operator A: U U, show that By the properties of the adjoit ad operator additio, A A is skew-adjoit. A A A A A A A A A A,,,,,,, ad therefore A A A A Give a liear operator A: U, A, A, A A, A A , so it is skew-adjoit. U, show that By the properties of the adjoit ad operator additio, ad therefore AA AA is self-adjoit. AA AA A A,,, A, A, A A, A A, AA AA, so it is self-adjoit

14 Show that if A is self adjoit, the A is self adjoit if ad oly if is real. A A A A A A, v, v, v, v, v, v ad A A if ad oly if which is tre if ad oly if is real. Show that if A is skew adjoit, the A is skew adjoit if ad oly if is real. A A A A A A, v, v, v, v, v, v ad A A if ad oly if which is tre if ad oly if is real. U U, show that A is the sm of a self-adjoit ad a skew-adjoit liear mappig. Becase A A A A A A A A A A A A A, ad A A 1 ad A A is skew adjoit. is self adjoit

15 Isometric liear operators Recall that a liear operator A: U R C U is isometric if ad oly if A for all vectors U. R, the A is a matrix with colms that form a orthoormal set, ad the matrix is called orthogoal. C, the A is a matrix with colms that form a orthoormal set, ad the matrix is called itary. Show that every isometric liear operator is ivertible. V V is isometric, the Av v for all vectors, bt if A is ot ivertible, there exists a o-zero v sch that Av 0. If this was tre, Av 0 0 v, as v was assmed to be o-zero. Ths, the matrix is ivertible. Show that A: U U is isometric if ad oly if 1 A A. By the properties of isometric liear operators ad the adjoit, A is isometric A A A, A,, AA, A A A Id 1 A V U U is isometric, show that A is isometric if ad oly if 1. Becase A is isometric, A ad ths, bt by the properties of the orm, A A A A if ad oly if 1. Corollary If the field associated with U is the reals, A is isometric if ad oly if A is isometric. is isometric, show that the rows of A also form a orthoormal set., As AA A A Id, the secod says that the colms of A form a orthoormal set, bt the first says that the cojgates of the rows of A form a orthoormal set, ad if the cojgates of the rows of A form a orthormal set, the so do the rows themselves.

16 Arge that a permtatio matrix is isometric. A T A = Id if A is iterpreted as a real matrix, ad if A is iterpreted as a complex matrix, as all the etries are real, A A = Id, so i either case, the iverse is the adjoit, i which case, it is isometric (orthogoal for real matrices ad itary for complex matrices).

17 Eigevales ad eigevectors, fid the dimesio ad a basis for the eigespace correspodig to a give ad kow eigevale. Give A ad, fid the row-eqivalet matrix ref Id A. The mber of free variables eqals the dimesio of the eigespace, ad to fid a basis for the eigespace, solve Id A 0, which is eqivalet to solvig ref Id A 0, which ca be solved sig backward sbstittio. Give a matrix, fid the eigevales. If a matrix is A: eigevales. is pper triaglar, lower triaglar or diagoal, the diagoal etries of the matrix are the If A is oe of these, there is a algorithm beyod the scope of this corse, the QR algorithm, that will fid eigevales i a merically stable maer. or two ad three dimesios, det Id A is always a polyomial i with a leadig term, ad ths is of degree. Sch a polyomial mst have complex roots, ad these roots are the eigevales. Show that if A A has a eigevale, that eigevale is real. Sppose AA for a o-zero vector. I this case,,,, A A, A A A, A A, ad therefore A ad as both orms are real, so mst. Show that if A is isometric ad has a eigevale, 1. Sppose A for a o-zero vector. By the properties of a orm ad isometric liear operators, A, ad therefore 1.

18 A liear operator A: is ot ivertible (that is, it is siglar) if ad oly if 0 is a eigevale of A. A is ot ivertible if ad oly if it is ot oe-to-oe, if ad oly if the ll space is ot jst {0 }, if ad oly if there is a o-zero vector sch that A 0 0, Give if ad oly if 0 is a eigevale of A. 1 1 A, show that A has o real eigevectors. 1 1 If A, Therefore, The agmeted matrix correspodig to this system of liear eqatios is ~ ~ Becase 1, it is always tre that the oly soltio to this is. Ths, the oly vector that 0 is a scalar mltiple of itself der mltiplicatio by A is the zero vector, ad ths A has o eigevales ad o eigevectors. Give 3 A, show that A has oly oe eigevector correspodig to the eigevale As 3 Id A 0, there is oe free variable v 1, so the dimesio of the eigespace is 1, ad a basis for this eigespace is fod by solvig this, amely, = 0 so eigevector correspodig to the eigevale , so 1 0 v is the sigle

19 A liear operator A: ca der certai circmstaces be writte as composed of the eigevectors of A. Describe how A VDV 1 operates? A VDV 1 where V is a matrix If the matrix A has liearly idepedet eigevectors, those eigevectors form a basis of. Coseqetly, we may write ay vector = Va. To fid A, we may either solve this system of liear eqatios, or we ca fid the iverse of V to get V 1 = a. The etry a k is the coefficiet of the k th eigevector v k. Mltiplyig Da mltiplies the k th etry by k, so the k th etry of DV 1 is a k k. Mltiplyig this vector by the matrix V calclates the liear combiatio of eigevectors k 1 a v, which is the reslt of mltiplyig A. k k k Uder which circmstaces ca we write A VDV 1 as A VDV? A symmetric matrix A: R R has real eigevales ad orthogoal eigevectors. We ca therefore ormalize these eigevectors ad prodce a orthogoal matrix V, so orthogoally diagoalizable, as V is a orthogoal matrix. A ormal matrix A: C A VDV T. We say that a symmetric matrix is C has complex eigevales ad orthogoal eigevectors. We ca therefore ormalize these eigevectors ad prodce a itary matrix V, so A VDV. We say that a ormal matrix is itarily diagoalizable, as V is a itary matrix. If we ca write A VDV, how ca we easily calclate 1000 A? By the properties of isometric liear operators (sqare matrices) ad operator compositio (matrix-matrix mltiplicatio), we have A As D is a diagoal matrix with etries 1000 VDV VDV VDV VDV VDV VDV times V D V V D V V DV V DIdDIdDV V DDD VD 999 times V times performed o a compter with a sigle fctio call. 999 times VDId DV D DV VD V V DV k, D 1000 is a diagoal matrix with etries 1000 k, a operatio that ca be

20 R calclate R is ot symmetric bt still diagoalizable, why may we have isses whe sig a similar techiqe to A VD V? If we are calclatig the iverse, withot more iformatio, this operatio may be merically stable.