MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I - Matrices, Vectors, Determinants, and Linear Algebra - Tadao ODA

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1 MATRICES, VECTORS, DETERMINANTS, AND LINEAR ALGEBRA Tadao Tohoku Uiversity, Japa Keywords: matrix, determiat, liear equatio, Cramer s rule, eigevalue, Jorda caoical form, symmetric matrix, vector space, liear map Cotets 1. Matrices, Vectors ad their Basic Operatios 1.1. Matrices 1.2. Vectors 1.3. Additio ad Scalar Multiplicatio of Matrices 1.4. Multiplicatio of Matrices 2. Determiats 2.1. Square Matrices 2.2. Determiats 2.3. Cofactors ad the Iverse Matrix 3. Systems of Liear Equatios 3.1. Liear Equatios 3.2. Cramer s Rule 3.3. Eigevalues of a Complex Square Matrix 3.4. Jorda Caoical Form 4. Symmetric Matrices ad Quadratic Forms 4.1. Real Symmetric Matrices ad Orthogoal Matrices 4.2. Hermitia Symmetric Matrices ad Uitary Matrices 5. Vector Spaces ad Liear Algebra 5.1. Vector spaces 5.2. Subspaces 5.3. Direct Sum of Vector Spaces 5.4. Liear Maps 5.5. Chage of Bases 5.6. Properties of Liear Maps 5.7. A System of Liear Equatios Revisited 5.8. Quotiet Vector Spaces 5.9. Dual Spaces Tesor Product of Vector Spaces Symmetric Product of a Vector Space Exterior Product of a Vector Space Glossary Bibliography Biographical Sketch Summary A dow-to-earth itroductio of matrices ad their basic operatios will be followed by

2 basic results o determiats, systems of liear equatios, eigevalues, real symmetric matrices ad complex Hermitia symmetric matrices. Abstract vector spaces ad liear maps will the be itroduced. The power ad merit of seemigly useless abstractio will make earlier results o matrices more trasparet ad easily uderstadable. Matrices ad liear algebra play importat roles i applicatios. Ufortuately, however, space limitatio prevets descriptio of algorithmic ad computatioal aspects of liear algebra idispesable to applicatios. The readers are referred to the refereces listed at the ed. 1. Matrices, Vectors ad their Basic Operatios 1.1. Matrices A matrix is a rectagular array a a a a j j 2 i1 i2 i m1 m2 mj m of etries a 11,, am, which are umbers or symbols. Very ofte, such a matrix will be deoted by a sigle letter such as A, thus a a a a j j 2 i1 i2 i m1 m2 mj m A :=. The otatio A = ( ) is used also, for short. I this otatio, the first idex i is called a the row idex, while the secod idex j is called the colum idex. Each of the horizotal arrays is called a row, thus ( a, a, a,, a, ),( a, a, a,, a, ),,( a, a, a,, a, ),,( a, a, a,, a, ) j j 2 i1 i2 i m1 m2 mj m are called the first row, secod row,, i -th row,, m -th row, respectively. O the other had, each of the vertical arrays is called a colum, thus

3 j 1 a a 21 a 22 2 j a 2,,,,, a a i1 a i2 a i a a m1 a m2 mj a m are called the first colum, secod colum,, j -th colum,, -th colum, respectively. Such a A is called a matrix with m rows ad colums, a ( m, )- matrix, or a m matrix. A ( m, )-matrix with all the etries 0 is called the zero matrix ad writte simply as 0, thus := Vectors A matrix with oly oe row, or oly oe colum is called a vector, thus ( a, a, a,, a, ) 1 2 is a row vector, while b 1 b 2 b i b m is a colum vector. j The rows ad colums of a ( m, )-matrix A above are thus called, the first row vector, secod row vector,, i -th row vector,, m -th row vector, ad the first colum vector, secod colum vector,, j -th colum vector,, -th colum vector. A (1, 1) -matrix, i.e., a umber or a symbol, is called a scalar Additio ad Scalar Multiplicatio of Matrices

4 The additio of two ( m, ) -matrices A = ( ) ad B = ( ) are defied by a b A+ B:= ( a + b ) = a + b a + b a + b a + b j 1 j 1 1 a21 b21 a22 b22 a2 j b2 j a2 b ai 1+ bi 1 ai2+ bi2 a + b ai + b i am1+ bm 1 am2+ bm2 amj + bmj am + bm whe the additio of the etries makes sese. The multiplicatio of a scalar c with a ( m, )-matrix A = ( ) is defied by ca := ( ca ) = a ca ca ca ca ca ca ca ca ca ca ca ca ca ca ca ca j j 2 i1 i2 i m1 m2 mj m whe the multiplicatio of a scalar with the etries makes sese Multiplicatio of Matrices What makes matrices most iterestig ad powerful is the multiplicatio, which does woders as explaied below. Suppose that the etries appearig i our matrices are umbers which admit multiplicatio. The the multiplicatio AB of two matrices A ad B is defied whe the umber of colums of A is the same as the umber of rows of B. Let A = ( a ) be a ( lm, ) -matrix ad B = ( b jk ) a ( m, ) -matrix. The their product is the ( l, ) -matrix defied by AB := ( c ), with c := a b, ik ik jk j= 1 m or more cocretely,

5 ab + + a b ab + + a b ab + + a b m m1 11 1k 1m mk m m i1 11 im m1 i1 1k im mk i1 1 im m l lm m1 l1 1k + + lm mk l lm m AB = ab + + a b ab + + a b ab + + a b. ab a b ab a b ab a b Of particular iterest is the product Av of a ( m, )-matrix A = ( ) with a colum vector v of size, which is the colum vector of size m defied by a a a a v a v + + a v 11 1 j ai1 a a i v j ai 1v1 aiv am1 amj am v am 1v1 amv + + = + +, as well as the product ua of a row vector u = ( u 1, u, m ) of size m with A, which is the row vector of size defied by ( u,, u,, u ) 1 i m a a a 11 1 j 1 i1 i m1 mj m a a a a a a = ( ua + + u a, ua, + + u a, ua, + + u a ) m m1 1 1j m mj 1 1 m m The traspose A T T A of a ( m), -matrix A = ( ) is the ( m, ) -matrix defied by a := ( a ), with a := a, ji ji or more cocretely, T a11 a12 a1 j a 1 a11 a21 ai1 am 1 a21 a22 a2 j a 2 a12 a22 ai2 a m2 := ai 1 ai 2 a a i a1 j a2 j a a mj am1 am 2 amj am a1 a2 ai a m.

6 For a ( lm, )-matrix A ad a ( m, )-matrix B, it is easy to see that T T T ( AB) = B A, whe the multiplicatio of the umbers cocered is commutative. Whe A ad B are (, ) -matrices, both products AB ad BA make sese, but they eed ot be the same i geeral. 2. Determiats 2.1. Square Matrices Square matrices, amely matrices with the same umber of rows ad colums, are most iterestig. Special amog them is the idetity matrix of size, deoted by I or I ad defied by 1 0 I = I := = ( δ), 0 1 where δ is kow as Kroecker s delta defied by 1 i = j δ :=. 0 i j For a arbitrary ( m, ) -matrix A, the followig clearly holds: I A= A ad AI = A. m The matrix ci with the same etry c alog the diagoal ad 0 elsewhere is called a scalar matrix. More geerally, a square matrix D = ( d ) of size is called a diagoal matrix if d = 0 for i j, that is, d d22 0 D =. 0 0 d 2.2. Determiats

7 Let A = ( ) be a square matrix of size (also said to be of order ), that is, a (, )- a matrix or a matrix. Whe the etries a are umbers (ratioal umbers, real umbers, complex umbers, or more geerally elemets of a commutative rig to be itroduced i Rigs ad Modules), for which additio, subtractio ad commutative multiplicatio are possible, associated to A is a umber called the determiat of A ad deoted by A or by det( A ). Whe = 1 or = 2, the determiat is defied to be a a a := a, := a a a a a21 a22 For = 3, the formula is a bit more complicated. a11 a12 a13 a21 a22 a23 := a11a22a33 + a12a23a31 + a13a21a32 a13a22a31 a11a23a32 a12a21a33. a a a The determiat of A = ( ) for geeral is defied as follows: 1 a a11 a1 := sg( σ ) a1 σ(1) a2 σ(2) ai σ( i) a σ( ), σ a a where σ rus through the permutatios of the idices {1, 2,,, }, ad sg( σ ) is the sigature of σ to be defied elsewhere i Groups ad Applicatios, sice it is ot so practical to compute the determiat usig this formula. Istead, there is a iductive way of computig the determiat: If how to compute the determiats of square matrices of size 1 is kow, the the determiat of a square matrix A of size is defied by j= 1 A := a Δ = a Δ + a Δ + + a Δ, 1 j 1 j where, for i ad j i geeral, Δ is the ( i, j) -cofactor of A defied by i+ j Δ :=( 1) det( A with the i-th row ad the j-th colum removed ). This formula is kow as the expasio of A with respect to the first row. I fact, it ca be show that the expasio with respect to the i -th row for ay i = 1,, gives rise to the same umber:

8 A = a Δ = a Δ + a Δ + + a Δ. j= 1 i1 i1 i2 i2 i i A similar formula holds whe the role of rows ad colums is iterchaged, that is, the expasio of A with respect to the j -th colum holds as well. I particular, A T = A. For square matrices A ad B of size, it ca be show that AB = A B Bibliography TO ACCESS ALL THE 29 PAGES OF THIS CHAPTER, Visit: Chateli, F. (1993): Eigevalues of Matrices. With exercises by Mario Ahués ad the author. Traslated from the Frech ad with additioal material by Walter Lederma. xviii+382 pp., Joh Wiley & Sos, Ltd., Chichester, ISBN [This is oe of the advaced books that deal with algorithmic ad computatioal aspects of matrices ad liear algebra.] Hogbert, L. ed. (2007). Hadbook of Liear Algebra, 1400 pp.,chapma & Hall/CRC, Boca Rato, Florida, ISBN [A complete treatmet of liear algebra.] Lag, S. (1989). Liear Algebra, Reprit of the Third Editio, x+285pp., Udergraduate Texts i Mathematics, Spriger-Verlag, New York, ISBN [A textbook iteded for a oe-term course at the juior or seior level. Origially published from Addiso-Wesley Publishig Compay.] Roma, S. (2005). Advaced Liear Algebra, Secod Editio, xvi+482 pp., Graduate Texts i Mathematics 135, Spriger-Verlag, New York, ISBN [A textbook iteded for advaced courses i liear algebra for graduate studets or advaced udergraduate studets.] Biographical Sketch Tadao, bor 1940 i Kyoto, Japa Educatio: BS i Mathematics, Kyoto Uiversity, Japa (March, 1962). MS i Mathematics, Kyoto Uiversity, Japa (March, 1964). Ph.D. i Mathematics, Harvard Uiversity, U.S.A. (Jue, 1967). Positios held: Assistat, Departmet of Mathematics, Nagoya Uiversity, Japa (April, 1964-July, 1968) Istructor, Departmet of Mathematics, Priceto Uiversity, U.S.A. (September, 1967-Jue, 1968) Assistat Professor, Departmet of Mathematics, Nagoya Uiversity, Japa (July, 1968-September, 1975) Professor, Mathematical Istitute, Tohoku Uiversity, Japa (October, 1975-March, 2003) Professor Emeritus, Tohoku Uiversity (April, 2003 to date)

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8 CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive