MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix C ab T is a matrix Prove that if > the Proof: Let C a 2 a a It is easy to see that rows of C are proportioal to b T with coefficiets a a that is a b T a 2 b T C a b T Therefore C by properties of ermiat 2 Let A M m are m matrices with m > Set C A T ; a it is a m m matrix Prove that C Proof: T is a m matrix where m > Therefore the system of homogeeous equatios T x has more variables tha equatios has a o-trivial solutio x ut the Cx ( A T ) x A ( t x ) A ( T ) A Hece C is a degeerate matrix C 3 I this problem we are workig with compoud 2 2 matrices of the form A C D where A C D M are matrices (a) Prove that O I ( ) ; I here I are the zero matrix the idetity matrix correspodigly of size (b) Express O A i terms of A Proof: (a) We kow that if we swap two differet rows i a ermiat the ermiat chages its sig Our matrix O I I is obtaied from the idetity matrix I 2 I I
MATH22 Liear Algebra Proof Problems 2 by movig each of first rows positios dowwards Obviously this requires 2 swaps of rows (b) Deote O I I I O ( ) 2 I ( ) 2 ( ) O I J I the O A O A A J A J A ( ) ( ) A by properties of ermiats of block-diagoal matrices 4 Let A be a 2 2 matrix which is similar oly to itself ot to ay other matrix Prove that A is a scalar matrix Proof: Sice A is similar oly to itself P AP A for all ivertible matrices P This is equivalet to AP P A Deote Take a b A P 2 The system of liear equatios yields Take a b 2 the the system yields [ 2 b c P a b a d Hece A is a scalar matrix [ a b a b Liear trasformatios Let T : V V be a liear trasformatio of a fiite dimesioal vector space V 5 Prove that if U < V is a vector subspace the T (U) { T (u) : u U } is a vector subspace of V 6 Prove that if T is oto it seds liearly idepedet sets of vectors to liearly idepedet sets 7 Prove that if T is ivertible U V the dim T (U) dim U 8 Let A be a matrix assume that A Prove that the map M M defied by X X A is a liear trasformatio 9 For give matrices A M 2 2 we defie the followig trasformatio R : M 2 2 M 2 2 R(X) AX Check that R is a liear trasformatio Prove that if R is oe-to-oe the both matrices A are ivertible Proof: Checkig the liearity of R is a direct expectio: R(X + Y ) A(X + Y ) (AX + AY ) AX + AY R(X) + R(Y ); R(cX) A(cX) (cax) cax cr(x) Assume that oe of the matrices A or is ot ivertible The it has ermiat
MATH22 Liear Algebra Proof Problems 3 the product AX has ermiat regardless of the choice of X Therefore all matrices i the image of trasformatio R have ermiat R caot be oto ut R is a liear trasformatio for liear trasformatios o fiite dimesioal vector spaces beig oto beig oe-to-oe are equivalet properties Hece R caot be oe-to-oe a cotradictio Aother solutio: Assume that R is oe-to-oe; as by part (ii) R is a liear trasformatio of a fiite-dimesioal vector space this is equivalet to assumig that R is oto Hece there is a matrix X such that AX I The A is ivertible with iverse X is ivertible with iverse AX Eigevalues eigevectors Prove that if a 2 2 matrix M has eigevalues the M ca be writte i the form a [c M d b for some umbers a b Hit: Use a matrix which cojugates M to a diagoal matrix Proof: M is cojugate to the diagoal matrix D say Deote the M A P DP p q P r s P P P P P [ s q r p s q p q r p r s qr qs pr ps [r s [ q p is i a desired form Aother solutio (shorter but more difficult): We start by showig that the claim is true for the most atural example of a matrix with eigevalues that is for D Ideed [ D Ay other such matrix M is similar to D so is equal to P DP hece M P DP P [ [ P It remais to deote by by P [ [ a b [ P [ Problem ca be stated i a much more geeral form solved without ivolvemet of eigevalues egevectors: Prove that if 2 2 matrix M is degeerate the M ca be writte i the form a [c M d b for some umbers a b Prove this statemet Symmetric orthogoal matrices 2 Recall that A are matrices the A A Prove that if A is a symmetric matrix is orthogoal the A is symmetric Proof: y defiitio of symmetric orthogo-
MATH22 Liear Algebra Proof Problems 4 al matrices Therefore A T A T ( A ) T ( A ) T ( T A ) T T A T ( T ) T T A A A Hece A is a symmetric matrix 3 A matrix A is atisymmetric if A T A Prove that if A is atisymmetric is a orthogoal matrix the A T is also atisymmetric Proof: y defiitio of atisymmetric orthogoal matrices Therefore A T A T ( A ) T ( A ) T ( T A ) T T A T ( T ) T T ( A) T A A ( A ) we ca suggest that the ermiat of a atisymmetric 3 3 matrix equals A proof is simple: we kow that that A A T for a atisymmetric matrix A this becomes A ( A) ( ) 3 A (we take out the scalar factor from each row of A that is three times!) therefore A A A (b) It clear that what matters i the proof above is that 3 is a odd umber So we wish to prove that It is easy: if is odd A is a atisymmetric matrix the A A A T for a atisymmetric matrix A this becomes A ( A) ( ) A (we take out the scalar factor from each of rows of A that is times!) Sice is odd ( ) hece A A A Hece A is a atisymmetric matrix 5 Prove that if a matrix is simultaeously triagular orthogoal the it is diagoal 4 Write several 3 3 atisymmetric matrices compute their ermiats (a) Make a cojecture about ermiats of atisymmetric 3 3 matrices prove it (b) Ca you geeralise your observatio to matrices of larger size prove it? Proof: (a) From a few simple examples such as 2 Proof: Assume that A is a upper triagular matrix of size that is all its etries below the diagoal equal (for lower triagular matrices the proof is the same with slight chage of words) Diagoal etries of A are its eigevalues; we kow that orthogoal matrices are ivertible; sice A is also orthogoal A is ivertible its eigevalues are ot equal We coclude that the diagoal etries a a 22 a are all o-zero Sice A is orthogoal its colums are orthogoal to each other The first
MATH22 Liear Algebra Proof Problems 5 colum of A is a a Colums a 2 a ca be orthogoal to a if their topmost etries a 2 a equal Repeatig the same argumet for a 2 we see that etries a 23 a 2 also equal Repeatig this process further we see that all elemets to the right of the diagoal equal ut the A is diagoal 6 Prove that the matrix 2 3 4 5 2 3 4 A 2 3 2 is ot similar to a symmetric matrix Ca A be similar to a atisymmetric matrix that is a matrix with the property A T A? Proof: Assume the cotrary let A is similar to a symmetric matrix S We kow that every symmetric matrix is similar (or cojugate) to a diagoal matrix; hece A is similar to some diagoal matrix D Sice A is triagular all eigevalues of A are diagoal etries equal Hece D is a diagoal matrix with all diagoal etries equal that is the idetity matrix ut the idetity matrix is similar oly to itself; that meas that A is the idetity matrix a obvious cotradictio A caot be similar to a atisymmetric matrix because A but atisymmetric matrices of odd size have zero ermiats