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# LINEAR ALGEBRA MATH 332

Colorado School of Mines

GPA 3.9

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This 3 page Class Notes was uploaded by Diana Prosacco on Monday October 5, 2015. The Class Notes belongs to MATH 332 at Colorado School of Mines taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/219617/math-332-colorado-school-of-mines in Mathematics (M) at Colorado School of Mines.

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Date Created: 10/05/15

David C Lay7 Linear Algebra and its A quot quot 3Ml ed Section NA pgs NA Lecture Basics of Matrices and Their Algebra Module 00 Suggested Problem Set NULL June 15 2009 Basic De nitions De nition Matrix A matrix is a set of elements organized by two indices into a rectangular array ln the case that these objects exist in the set of complex numbers we write A6 men where nm E N1 At the element level we have that an a12 dis v aln a21 a22 a23 v a2n A an M2 ass aSn where Alijaij aijEC fori123 m andj123 n 1 aml amZ amS amn 0 ln the case that n m we call the matrix square Otherwise it is called rectangular o For a square matrix the entries running from the upper left to the lower right are called the main diagonal entries De nition Vector A column vector or just vector is matrix of size n X l where n E N A row vector is matrix of size l X n where n E N At the element level we have that U1 U2 v US where vi EC foril23n 2 v71 r r1 r2 r3 r7 whererjECforjl23n 3 De nition Scalar A scalar is a matrix whose size is l X 1 ln this case that this scalar is an object from the real numbers we write a E R De nition Equality of Matrices Two matrices AB E men are said to be equal if and only if aij bij for i 123m andj 123n Unitary Operations De nition Transposition Given A E Rmxn we de ne the transpose of A to be the matrix AT 6 Rnxm such that an a21 asi v ami a12 a22 a32 v am2 AT 0413 a23 ass u ams 4 aln 271 067 amn o If A is such that A AT then the matrix A is called symmetric 2 1Often it is useful to consider elements that are functions However it is traditional and straightforward to rst consider matrices of numbers 2It can be shown that the eigenvalues of symmetric matrices are always real numbers Basics of Matrices and Their Algebra OOl o If A is such that AT 7A then the matrix A is called skewsymmetric 3 0 Using the previous de nitions one can quickly show that A BT AT BT assuming that the matrices are such that their addition is wellde ned 4 De nition Conjugation Given A E men de ne the conjugate of A to be the matrix A E men such that amp11 amp12 a13 i ampin amp21 amp22 a23 i i 271 A amp31 amp32 533 u 371 5 ami am2 ms i mn o The bar implies complex conjugation That is if c E C then c a bi a b E R and E a 7 hi De nition Adjoint Given A E men de ne the adjoint or Hermitian of A to be the matrix AH E men such that AH AT AT 5 o The adjoint is considered as an extension of the transpose to matrices with complex numbers Sometimes the adjoint is called the Hermitian of a matrix 0 A matrix is called selfadjoint if AH A 5 o A matrix is called skewadjoint if AH 7A 7 Binary Operations De nition Addition and Scalar Multiplication of Matrices Let A B E men then A B C is de ned such that CE Cnxm where 017 aij by for i 123 m and j 123 n Also let 8 E C then SA C where 017 s aij for i 123m andj 123n From these de nitions we have the general properties for addition and scalar multiplication of matrices 1 A B B A 2 ABC ABC 3 A 0 A 4 A 71 A 0 where 0 denotes an m X n matrix whose elements are the scalar zero 5 TABTATB 53gt T3ATA3A 7 7 SA 7 8A 8 l A A where ABCE men and 7 S E C 3It can be shown that the eigenvalues of skewesymmetric matrices are always imaginary numbers or the number zero 4F rom this it follows that a matrix can always be written as the sum of a symmetric and skewesymmetric matrix To show this note that A A AT A 7 AT 5It is often the case that the Hermitian is denoted All 6It can be shown that the eigenvalues of selfeadjoint matrices are always real numbers 7It can be shown that the eigenvalues of skeweadjoint matrices are always imaginary numbers or the number zero Basics of Matrices and Their Algebra 002 De nition Matrix Product Lei A E men and B E Cpxq If n p then AB C is de ned such that CE meq where cij Zaikbkj The general properties for matrix products are k 1 1 ABC ABC 2 ABC ABAC 3 BCA BACA 4 MAB 7 AB ATB 5 ImA A ALL where A B C are de ned appropriately and 7 E C o It is not necessarily the case that ABBA That is matrix multiplication does not in general commute o The identity matrix Ik is a square matrix with the scalar identity ie the number one on the main diagonal That is Ikxklij 1 ifi j and Ikxklij 0 ifi j o The inverse matrix of a square matrix A is the square matrix A 1 such that AA 1 A IA I De nition lnner Product Given XE Rn and ye Rn de ne the inner product of x and y to be n X39 XT 6 i1 n 0 Using the inner product it is possible to de ne matrix multiplication as cij Z aikbkj aibj 161 where ai is the ith row of A and bj is the jth column of B 0 When working with complex vectors then it is typical to de ne the inner product to be xHy It is rare to multiply matrices with this de nition De nition Outer Product Given XE Rn and ye Rn de ne the outer product of x and y to be xyT It is easily veri ed that this product results in an n X n matrix 0 If we take on faith that ABT BTAT then we can also see that the outer product produces a symmetric matrix 3 0 When working with complex vectors then it is typical to de ne the outer product to be Ky 8To prove the aforementioned equality note that ABij ai bj thus the Ljielement of the transpose of AB is 3739 bi which is the product of the jthirow of A and ithicolulnn of B Since the ithicolulnn of B is the ithirow of BT and the jthirow of A is the jthicolulnn of AT the desired equality follows Basics of Matrices and Their Algebra 00 3

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