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# MATH BIOPHYSICS MAP 5485

FSU

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This 19 page Class Notes was uploaded by Tremaine Johnson on Thursday September 17, 2015. The Class Notes belongs to MAP 5485 at Florida State University taught by John Quine in Fall. Since its upload, it has received 41 views. For similar materials see /class/205470/map-5485-florida-state-university in Applied Mathematics at Florida State University.

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Date Created: 09/17/15

A VERY BRIEF LINEAR ALGEBRA REVIEW 5485 Introduction to Mathematical Biophysics Fall 2006 Introduction Linear Algebra also known as matrix theory is an important el ement of all branches of physics and mathematics Very often in this course we study of the shapes and the symmetries of molecules Motion of 3D space which leave molecules rigid can be described by matricesi Brie y mentioned in these notes will be quantum mechanics where matrices and their eigenvalues have an essential ro e In all cases it is useful to allow the entries in the matrix to be complex numbers If you have studied matrices only with real number entries it is very easy to adapt to complex numbers Almost all the rules are the same In doing computations with matrices it is useful to have a computer program such as Maple or Matlabi These tools make multiplication of matrices very easy and they work with complex numbers The main difference between Maple and Matlab is that Maple can work symbolically that is you can use letter as well as numbers for entries When using numbers Matlab is often fasteri Below we give a review of a few basic ideas that will be used in the course 2 1 A is a matrix with 2 rows and 2 columns ie a 2 X 2 matrix A matrix with m rows and n columns is called an m X n matrix A matrix with the same number of rows and columns is called a square matrix 3 X 3 square matrix Matrices Example 3 1 7 B 71 2 0 0 1 5 3 X 2 matrix 2 0 C 79 10 1 14 A 1 X 1 matrix is the same as a number or scalar 3 Vectors Matrices with 1 row are called row vectors and matrices with 1 column are called column vectors are column VeCtOI Si 2 are row vectors Usually we will assume vectors are column vectors A row vector can be converted into a column vector or vice versa by the transpose operation Which changes rows to columns Example 1 1 71 0t 71 0 Complex Numbers Complex numbers can be used in matrices A number 2 a bi Where a and b are real numbers and Where i fl is called a complex number The number a is called the real part a 992 and b the imaginary part b 32 If 2 a bi Where a and b are real numbers then the complex conjugate of 2 is E a 7 bi The number 2 is real if b 0 or equivalently 2 2 Basic operations With matrices are 2 0 addition 0 scalar multiplication o multiplication Matrix Addition Add matrices by adding corresponding entries 1 11 2112 1 Scalar Multiplication Every matrix entry is multiplied by the scalar 2i2 1 4i 2 fl 3 722 62 Matrix Multiplication Multiplication AB can be done only if the number of columns of A is the same as the number of rows of B Each entry of the product is the dot product of a row of the rst matrix With a column of the second 2 13 1 211032 8 7112 0 7111022 3 3 11 2 311012 5 1 3 1 71 0 1 0 1 1 2 g 2 1 0 3 0 1 211133 2711230 201 31 1101131710210100 11 211103 2711200 201 01 12 0 35 4 71 1 3 0 5 An important thing to remember about matrix multiplication is that it is not commutative in general AB BA For example lt11gtlt19gt lt 1gt 1 0 1 1 7 1 1 1 1 0 1 7 1 2 Other operations are 0 conjugation o transpose 0 adj oint Conjugate 1f Z is a matrix then the matrix conjugate is formed by taking the complex conjugate of each entry ExampleLet 71i 2 71 2 1 0 12H ng 114 212 ABi Where 1 2 1 1 3 and 1 0 311 1 The matrix A is the real part of Z and the matrix B is the imaginary part of Z The conjugate of Z is 7 1 7 i 2 Z l3 i 7 ml or using the real and imaginary parts of the matrix7 ZA7Bi Properties of the matrix congugate BX ABXE Transpose The transpose of a matrix A7 At7 is obtained by changing rows to columns or equivalently7 changing columns to rows1 Sometimes the transpose is denoted A rather than Ah 2 1 71 2 0 71 A 0 1 2 AL 1 1 71 0 1 71 2 1 Properties of transpose AB B A AB A B 4 Adjoint The Hermitz39zm transpose or odjoz39nt is the conjugate transpose given by 7t A Ai Example 1 127i3 2i 3 Example For the matrix Z given above 7 it 7 1 7 239 3 239 Z Z l 2 7 221 Properties of Adjoiut ABY BA AB A B a z Dot product Let 1 b and w y be two vectors then the dot product is c 2 given by z vwv w a b c y azbyc2 2 1 z 2 Let 1 1 and w 1 7i then the Hermz39tz39zm dot product is given by i 3 ltvwgt vw iw 1 7 i21 1 7 7i3 3 7 6 it The length of a vector lvl is given by 2 l1 lt1 vgtl Two vectors 1 and w are orthogonal or perpendicular if ltvwgt 0 In general for real vectors ltvwgt cost Where 9 is the angle between the vectors Matrix multiplication and dot product An m X n matrix can be considered as a list of n X 1 column vectors Alv17m7vnl and the transpose as a list of row vectors At B i wt is a k X in matrix given as row vectors7 then wlvl wlvn BA wkvl wkvn is a matrix of dot products Symmetric Matrices A matrix B is symmetric if B Bt Example 3 1 i 2 1 i 0 75 2 75 2 is symmetric A matrix B is self adjoint or Hermitian symmetric if Bquot B Example 2 1i 2i l7i 3 5 72i 5 is self adjoint Determinant The determinant of a 2 X 2 matrix is given by det Z Z Z ad7bc Fora3gtlt3matrix7 a b c detd e f a 7b d cdz g h k g g This is called expansion by minors Likewise the determinant is de ned for any square matrix Properties of determinant detAB detA detB detA 0 implies A71 exists Vector Cross Product a z 122 7 cy b X y 7a2 01 c 2 ay 7 121 b2 7 cyi 7 a2 7 czj ay 7 bzk Where 1 0 0 10 j1 k0 0 0 1 The formula for cross product is often remembered by pretending that i7 j and k are numbers and Writing a z i j k b X y det a b c c z z y 2 Identity Matrix The matrix 1 denotes a square matrix Whose entries are aij were 1ifz39j av Oifiy ji The matrix 1 is called the unit or identity matrix Identity matrices come in different sizes 1 1 0 13 0 l0 1 0 Matrix inverse Let A be a square matrix H detA 0 there is an inverse A71 such that AilA AA 1 I The inverse of a 2 X 2 matrix is easy to nd If A an an 0421 0422 I2 OHO 0 0 1 then A71 l a22 a12 a 7amp21 all Where a anagg 7 algagl is the determinant of A1 Eigenvalue and Eigenvectors The scalar A a real or complex number7 is an eigenvalue of a matrix A corresponding to an eigenvector v f 0 if Av Av 11 11113111 The eigenvalue is 37 and an eigenvector is 11 1 Note that 22t is also an 1 311 1321 111 21321 xample 2 The eigenvalue is i and an eigenvector is 17 filti Example 3 0 71 1 7 7239 7 1 1 0 239 1 1 239 The eigenvalue is 7i 7 and an eigenvector is Lil i Note that the equation in example 3 is the conjugate on the one in example 21 Also note that a matrix With real entries can have complex eigenvalues and eigenvectorsi Eigenvalues of self adjoint matrices are real This fact is essential in many areas of mathematics and is also a key fact in the mathematical formulation of quantum mechanics Here is a proof If A is self adjoint and A1 Av then taking the adjoint of both sides gives v A Xvi Multiplying the first equation on the left by 1 and the second on the right by 1 gives 7 v Av A U Q AM Since v f 0 we have X A and so A is real A consequence is that the eigenvalues of a symmetric real matrix are real Rotation matrices in dimension 2 rotation of a column vector by an angle 9 counterclockwise is given by multiplying on the left by the matrix cos 9 7 sin 9 sin 9 cos 9 in dimension 3 rotations about the three axes are given by 0 Rotation an angle 9 about the z axis 1 0 0 R109 0 3086 7sint9 0 sin 9 cos 9 o Rotation an angle 9 about the y axis cos 9 0 sint9 Ry9 0 l 0 7 sin 9 0 cos 9 o Rotation an angle 9 about the 2 axis cos 9 7 sin 9 0 R209 sint9 3086 0 0 0 l Matrices can be found for rotation of any angle about axis7 Where an axis is given by a direction vector of length onei A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2006 Introduction Linear Algebra also known as matrix theory is an important element of all branches of physics and mathemat ics Very often in this course we study of the shapes and the symmetries of molecules Motion of 3D space which leave molecules rigid can be described by matrices Brie y mentioned in these notes will be quantum mechanics where matrices and their eigen values have an essential role In all cases it is useful to allow the entries in the matrix to be complex numbers If you have studied matrices only with real number entries it is very easy to adapt to complex numbers Almost all the rules are the same In doing computations with matrices it is useful to have a computer program such as Maple or Matlab These tools make mul tiplication of matrices very easy and they 2 work with complex numbers The main dif ference between Maple and Matlab is that Maple can work symbolically that is you can use letter as well as numbers for en tries When using numbers M atlab is often faster Below we give a review of a few basic ideas that will be used in the course Matrices Example 2 1 A l 1 l A is a matrix with 2 rows and 2 columns ie a 2 X 2 matrix A matrix with m rows and 71 columns is called an m X 71 matrix A matrix with the same number of rows and columns is called a square matrix 3 X 3 square matrix 317 B 120 015 3 X 2 matrix 2 0 C 9 1O 1 14 A 1 X 1 matrix is the same as a number or scalar 3 3 Vectors Matrices with 1 row are called row vectors and matrices with 1 column are called column vectors are column VGCtOI S C21 D321 are row vectors Usually we will assume vec tors are column vectors A row vector can be converted into a column vector or vice versa by the transpose operation which changes rows to columns Example 1 1 1 of 1 0 Complex Numbers Complex numbers can be used in matrices A number 2 a bi Where a and b are real numbers and Where Z2 1 is called a complex number The number a is called the real part a 3 and b the imaginary part b 2 If 2 a bi Where a and b are real num bers then the complex conjugate of z is Z a bi The number 2 is real if b 0 or equivalently z 2 Basic operations With matrices are 0 addition 0 scalar multiplication o multiplication Matrix Addition Add matrices by adding corresponding entries iiHiSlW l 5 Scalar Multiplication Every matrix en try is multiplied by the scalar 221 42 22 1 3 2i 62 Matrix Multiplication Multiplication AB can be done only if the number of columns of A is the same as the number of rows of B Each entry of the product is the dot product of a row of the rst matrix with a column of the second 31 3111193322 21 1 2J F gt J a 3ll0l2 gt l3gtd O 211133 2711230 2 01 31 110113 1710210 10011 211103 2711200 201 01 12 0 35 4 l 1 3 0 5 6 An important thing to remember about matrix multiplication is that it is not com mutative in general AB y BA For exam mum lt12gtltsigtlt Other operations are c conjugation o transpose o adjoint Conjugate If Z is a matrix then the ma trix conjugate is formed by taking the com plex conjugate of each entry Example Let 71z 2 712 10 Z3 i2i3 12Z ABZ 12 Alwl Where 1 0 B l 1 2l The matrix A is the real part of Z and the matrix B is the imaginary part of Z The conjugate of Z is 1 z 2 Z3r m or using the real and imaginary parts of the matrix ZA m Properties of the matrix eongugate KE ABKE I ranspose The transpose of a matrix A At is obtained by Changing rows to columns or equivalently Changing columns to rows Sometimes the transpose is denoted A rather than At 1 1 12 At 1 gt D 2 A 0 1 Di O Fol Properttes of transpose ABY 13W ABt AtBt Adjoint The Hermtttcm transpose 01 ad jomt is the conjugate transpose given by Aquot At Example 1 12 z 3 2t 3 Example For the matrix Z given above 7771 t3t Z Z 2 22 Properttes of Adjotnt AB BA AB AB Dot product Let 1 b and w x y be two vectors then the dot product 2 is given by vwvtw a b C axbycz z 1Z 2 Let U 1 and w 1 Z then Z 3 the Hermitian dot product is given by ltvwgt vw Ww 1 i211 03 3 ii The length of a vector M is given by M2 lt1 vgt Two vectors 1 and w are orthogonal or perpendicular if U wgt 0 In general for real vectors ltvwgt cos6 Where 8 is the angle between the vectors Matrix multiplication and dot prod uct An m X 71 matrix can be considered as a list of n X 1 column vectors A v1vn and the transpose as a list of row vectors it W lvzl If B wt is a k X m matrix given as row vectors then w1v1 w1vn BA s s wkU1 when is a matrix of dot products Symmetric Matrices A matrix Bis sym metric if B Bt Example 3 1i 2 1i 0 5 l2 5 2 is symmetric A matrix B is self adjoint 0r Hermitian symmetric if 13 13 Example 2 1i 2i 1 i 3 5 2i 5 4 is self adjoint Determinant The determinant of a 2 X 2 matrix is given by ab ab detL d Cdad bc Fora3gtlt3matrix abc detldefla2 bd cdz gym 9 9 This is called expansion by minors Like wise the determinant is de ned for any square matrix

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