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Matrices and Tensors APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM... M ¡°11 12 1c ¡°MM... M M = ¡°21 22 2c ¡°. (A1) ¡°...... ¡°MM.... ¢±¡°rrc1 The typical element of the array, Mij, is the ith element in the jth column; in this text elements Mij will be real numbers or functions whose values are real numbers. The transpose of matrix M is denoted by MT and is obtained from M by interchanging the rows and columns: ¯MM... M ¡°11 21r 1 ¡°MM... M MT = ¡°12 22r 2 ¡°. (A2) ¡°. ... ¡°MM. ... ¢±¡°1crc 595 596 APPENDIX:MATRICES AND TENSORS The operation of obtaining MT from M is called transposition. In this text we are interested in special cases of r-by-c matrix M. These special cases are those of the square matrix, r = c = n, the case of the row matrix, r =1, c = n, and the case of the column matrix, r = n, c = 1. Fur- ther, the special subcases of interest are n = 2, n = 3, and n = 6; subcase n = 1 reduces all three special cases to the trivial situation of a single number or scalar. Square matrix A has the form ¯A AA... ¡°11 12 1n ¡°A AA... A = ¡°21 22 2n ¡°, (A3) ¡°...... ¡°A ....A ¢±¡°nnn1 while row and column matrices r and c have the forms ¯c ¡°1 ¡°c ¡°2 ¡°. r = [rr r] c = ¡° 12... n , ¡°, (A4) ¡°. ¡° ¡°. ¡° c ¢±¡°n respectively. The transpose of a column matrix is a row matrix, and thus T = [ ] c cc12... cn . (A5) To save space in books and papers, the form of c in (A5) is used more frequently than the form in the second of (A4). Wherever possible, square matrices will be denoted by upper-case bold- face Latin letters, while row and column matrices will be denoted by lower-case boldface Latin letters, as is the case in eqs. (A3) and (A4). A.3. THE TYPES AND ALGEBRA OF SQUARE MATRICES The elements of square matrix A given by (A3) for which the row and column indices are equal, namely elements A11, A22, … , Ann, are called diagonal elements. A matrix with only di- agonal elements is called a diagonal matrix: ¯A 0...0 ¡°11 ¡°0...0A A = ¡°22 ¡°. (A6) ¡°...... ¡°0....A ¢±¡°nn The sum of the diagonal elements of a matrix is a scalar called the trace of the matrix and, for matrix A, it is denoted by trA: =+++ trA A11AA 22 ... nn . (A7) If the trace of a matrix is zero, the matrix is said to be traceless. Note also that trA = trAT. TISSUE MECHANICS 597 The zero and the unit matrix, 0 and 1, respectively, constitute the null element, the 0, and the unit element, the 1, in the algebra of square matrices. The zero matrix is a matrix whose every element is zero and the unit matrix is a diagonal matrix whose diagonal elements are all one: ¯00...0 ¯10...0 ¡°¡° ¡°00...0 ¡°01...0 0 = ¡°, 1 = ¡°. (A8) ¡°¡° ¡°...... ¡°...... ¡°¡° ¢±¡°0....0 ¢±¡°0....1 δ A special symbol, the Kronecker delta, ij, is introduced to represent the components of the unit δ δ δ ≠ matrix. When i = j the value of the Kronecker delta is 1, 11 = 22 = … = nn = 1, and when i j δ δ δ δ the value of the Kronecker delta is 0, 12 = 21 = … = n1 = 1n = 0. Multiplication of matrix A by a scalar is defined as multiplication of every element of matrix A by scalar α; thus, ¯ααA AA... α ¡°11 12 1n ¡°ααA AA... α αA ¡°21 22 2n w ¡°. (A9) ¡°...... ¡°ααA ....A ¢±¡°nnn1 It is then easy to show that 1A = A, –1A = –A, 0A = 0, and αO = 0.The addition of square matrices is defined only for matrices with the same number of rows (or columns). The sum of two matrices, A and B, is denoted by A + B, where ¯A ++BAB... AB + ¡°11 11 12 12 1nn 1 ¡°A ++BAB... A + B AB+ ¡°21 21 22 22 2nn 2 w ¡°. (A10) ¡°...... ¡°A ++BAB.... ¢±¡°n11 n nn nn Matrix addition is commutative and associative, ABBA+=+ and ABC++=()() ABC ++, (A11) respectively. The following distributive laws connect matrix addition and matrix multiplication by scalars: ααα()AB+= A + B and ()αβ+=+AAA α β, (A12) where α and β are scalars. Negative square matrices may be created by employing the defini- tion of matrix multiplication by scalar (A8) in the special case when α = –1. In this case the definition of addition of square matrices (A10) can be extended to include subtraction of square matrices, A – B. A matrix for which B = BT is said to be a symmetric matrix, while a matrix for which C = –CT is said to be a skew-symmetric or anti-symmetric matrix. The symmetric and skew- symmetric parts of a matrix, say A, are constructed from A as follows: ¬1­ T symmetric part of AAAw ­()+ , and (A13) ®2­ ¬1­ T skew-symmetric part of AAAw ­(). (A14) ®2­ 598 APPENDIX:MATRICES AND TENSORS It is easy to verify that the symmetric part of A is a symmetric matrix and that the skew- symmetric part of A is a skew-symmetric matrix. The sum of the symmetric part of A and the skew-symmetric part of A is A: ¬11­­TT ¬ AAAAA=++­­()() . (A15) ®22­­ ® This result shows that any square matrix can be decomposed into the sum of a symmetric and a skew-symmetric matrix. Using the trace operation introduced above, representation (A15) can be extended to three-way decomposition of matrix A: (trA ) ¬11­­ ¯T ¬(trA ) ¬ T A1=+­­¡°()2)AA+ ­ +() AA. (A16) ­­¡° ­ n ®22¢±¡°®n ® The last term in this decomposition is still the skew-symmetric part of the matrix. The second term is the traceless symmetric part of the matrix, and the first term is simply the trace of the matrix multiplied by the unit matrix. Example A.3.1 Construct the three-way decomposition of matrix A given by ¯ ¡°123 ¡° A = ¡°456. ¡° ¢±¡°789 Solution: The symmetric and skew-symmetric parts of A, as well as the trace of A are calculated: ¯ ¯ ¡°135 ¡ 0 1 2 ° ¬11­­+=TT¡° ¬ = ¡ °= ­­()357,AA¡° ()101, AA ¡ ° tr15 A ; ®22¡° ® ¡ ° ¢±¡°579 ¢¡ 2 1 0 ±° then, since n = 3, it follows from (A16) that ¯ ¯ ¯ ¡°¡°¡°500 435 0 1 2 ¡°¡°¡° A =+¡°¡°¡°050 3 07 + 1 0 1. ¡°¡°¡° ¢±¢±¢±¡°¡°¡°005 5 74 2 1 0 Introducing the notation for the deviatoric part of n-by-n square matrix A, (trA ) devAA= 1, (A17) n the representation for matrix A given by (A16) may be rewritten as AHDS=++, (A18) where H is called the hydrostatic component, D is called the deviatoric component, and S is the skew-symmetric component, (trA ) ¬ ¬ = =+1­ T = 1­ T H1, DAA ­(dev dev ) , SAA ­() . (A19) n ®2­ ®2­ TISSUE MECHANICS 599 Example A.3.2 Show that tr(devA) = 0. Solution: Applying the trace operation to both sides of (A17), one obtains tr(devA) = trA – (1/n)trA tr1; then, since tr1 = n, it follows that tr(devA) = 0. The product of two square matrices, A and B, with equal numbers of rows (columns) is a square matrix with the same number of rows (columns). The matrix product is written as A⋅B where A⋅B is defined by kn= ()AB = A B ; (A20) ¸ ij ik kj k=1 thus, for example, the element in the rth row and cth column of product A⋅B is given by =+++ (AB¸ )rcA r11BAB c r 2 2 c ... AB rn nc . The dot inside matrix product A⋅B indicates that one index from A and one index from B are to be summed over. The positioning of the summation index on the two matrices involved in a matrix product is critical and is reflected in the matrix notation by the transpose. In the three equations below, (A21), study carefully how the positions of the summation indices within the summation sign change in relation to the position of the transpose on the matrices in the asso- ciated matrix product: kn== kn kn = ()ABTT= A BABAB , () A B= , ( A TT B )= . (A21) ¸ ij ik jk¸ ij ki kj¸ ij ki jk kk==11 k = 1 A widely used notational convention, called the Einstein summation convention, drops the summation symbol in (A20) and writes = ()AB¸ ijA ikB kj , (A22) where the convention is the understanding that the repeated index, k, is to be summed over its range of admissible values from 1 to n.
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