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INTRODUCTION to MATRIX ALGEBRA 1.1. Definition of a Matrix INTRODUCTION TO MATRIX ALGEBRA 1. DEFINITION OF A MATRIX AND A VECTOR 1.1. Definition of a matrix. A matrix is a rectangular array of numbers arranged into rows and columns. It is written as a11 a12 ... a1n a a ... a 21 22 2n .. (1) .. .. am1 am2 ... amn The above array is called an m by n (m x n) matrix since it has m rows and n columns. Typically upper-case letters are used to denote a matrix and lower case letters with subscripts the elements. The matrix A is also often denoted A = kaijk (2) 1.2. Definition of a vector. A vector is a n-tuple of numbers. In R2 a vector would be an ordered 3 n pair of numbers {x, y}.InR a vector is a 3-tuple, i.e., {x1,x2,x3}. Similarly for R . Vectors are usually denoted by lower case letters such as a or b, or more formally ~a or ~b. 1.3. Row and column vectors. 1.3.1. Row vector. A matrix with one row and n columns (1xn) is called a row vector. It is usually written ~x 0 or 0 ~x = x1,x2,x3, ..., xn (3) The use of the prime 0 symbol indicates we are writing the n-tuple horizontally as if it were the row of a matrix. Note that each row of a matrix is a row vector. 1.3.2. Column vector. A matrix with one column and n rows (nx1) is called a column vector. It is written as x1 x 2 x3 ~x = . (4) . . xn Note that each column of a matrix is a column vector. It is common to write the columns of a matrix as a1,a2, ...an where each column vector aj is of length m. As an example a2 is given by Date: August 27, 2004. 1 2 INTRODUCTION TO MATRIX ALGEBRA a12 a 22 a32 ~a 2 = . (5) . . am2 2. VARIOUS TYPES OF MATRICES AND VECTORS 2.1. Square matrices. A square matrix is a matrix with an equal number of rows and columns, i.e. m=n. 2.2. Transpose of a matrix. The transpose of a matrix A is a matrix formed from A by interchanging rows and columns such that row i of A becomes column i of the transposed matrix. The transpose is denoted by A0 or AT and 0 A = kajik when A = kaijk (6) 0 0 0 If aij is the ijth element of A , then aij = aji . If the matrix A is given by 32 5 7 14 6 3 A = (7) 510−20 1115−2 then A0 is given by 31 5 1 24 101 A0 = (8) 56−215 73 0 −2 2.3. Symmetric matrix. A symmetric matrix is a square matrix A for which A = A0 (9) An example of a symmetric matrix is 31 5 1 14101 T = (10) 510−215 1115−2 31 5 1 14101 T 0 = 510−215 1115−2 INTRODUCTION TO MATRIX ALGEBRA 3 2.4. Identity matrix. The identity matrix of order n written I or In, is a square matrix having ones along the main diagonal (the diagonal running from upper left to lower right and zeroes elsewhere). 100... 0 010... 0 001... 0 .. (11) .. .. 000... 1 If we write I = kδijk then ( 1,i= j δij = (12) 0,i=6 j The symbol δij is called the Kronecker delta. Note that for a system of n equations in n unknowns that has a unique solution, the coefficient matrix of the system after performing the appropriate number of row and column operations is an identity matrix. 2.5. Scalar matrix. For any scalar λ, the square matrix S = k λδij k = λI (13) is called a scalar matrix. An example is 3000 0300 (14) 0030 0003 2.6. Diagonal matrix. A square matrix D = k λi δij k (15) is called a diagonal matrix. Notice that λi varies with i. An example is 13 0 0 0 02 0 0 (16) 00−40 00 0 56 If a system of equations was written with this coefficient matrix, we could solve the system by solving each equation individually. 2.7. Null or zero matrix. The null or zero matrix is a matrix with each element being zero. It is denoted as 0. 4 INTRODUCTION TO MATRIX ALGEBRA 000... 0 000... 0 000... 0 0 = .. (17) .. .. 000... 0 2.8. Upper triangular matrix. A matrix with all elements below the main diagonal equal to zero is called an upper triangular matrix. a11 a12 a13 ... a1n 0 a a ... a 22 23 2n 00a33 ... a3n A = .. (18) .. .. 000... amn Specifically aij =0if i>jas long as i<mand j<n. 2.9. Lower triangular matrix. A matrix with all elements above the main diagonal equal to zero is called a lower triangular matrix. a11 00... 0 a a 0 ... 0 21 22 a31 a32 a33 ... 0 A = .. (19) .. .. am1 am2 am3 ... amn Specifically aij =0if i<jas long as i<mand j<n. 3. A NOTE ON SUMMATION NOTATION 3.1. Single sums. 3.1.1. Definition of a single sum. n X ai = am + am+1 + am+2 + ... + an (20) i=m 3.1.2. Properties of a single sum. n n X kai = k X ai (21) i=1 i=1 n X k = k + k + k + ... + k = nk i=1 n n n X (ai + bi )=X ai + X bi i=1 i=1 i=1 INTRODUCTION TO MATRIX ALGEBRA 5 3.2. Double sums. 3.2.1. Definition of a double sum. n m m m m X X aij = X a1j + X a2j + ... + X anj (22) i=1 j=1 j=1 j=1 j=1 =a11 + a12 + a13 + ... + a1m +a21 + a22 + a23 + ... + a2m +... +... +... +an1 + an2 + an3 + ... + anm 3.2.2. Properties of a double sum. n n n 2 (X aj )(X ai )= X ai +2XXaiaj (23) j=1 i=1 i=1 i<j n 2 = X ai + XXaiaj i=1 i6=j 6 INTRODUCTION TO MATRIX ALGEBRA 4. MATRIX OPERATIONS 4.1. Scalar multiplication (matrix). Given a matrix A and a scalar λ, the product of λ and A, writ- ten λA, is defined to be λa11 λa12 ... λa1n λa λa ... λa 21 22 2n .. λA = (24) .. .. λam1 λam2 ... λamn 4.2. Scalar multiplication (vector). Given a column vector ~a and a scalar λ, the product of λ and ~a , written λ~a, is defined to be λa1 λa 2 . λ~a = (25) . . λam For the second column of a matrix we could write λa12 λa 22 λa32 λ~a2 = . (26) . . λam2 4.3. Trace of a square matrix. The trace of a matrix is the sum of the diagonal elements and is denoted tr A. Consider the matrix C below. 31 5 1 14101 C = (27) 510−215 1115−2 The trace of C is [3 + 4 + -2 + -2] = 3. 4.4. Addition of vectors. - The sum c of a vector a with m elements and a vector b having m elements is a vector with m elements and whose elements are given by cj = aj + bj ∀ j (28) This gives INTRODUCTION TO MATRIX ALGEBRA 7 c1 a1 b1 a1 + b1 c a b a + b 2 2 2 2 2 . . . . ~c = = + = (29) . . . . . . . . cm am bm am + bm 4.5. Linear combinations of vectors. If a and b are two n-vectors and s and t are two real numbers, tz + sb is said to be the linear combination of a and b. In symbols we write, a1 b1 ta1 + sb1 a b ta + sb 2 2 2 2 . . . t + s = (30) . . . . . . am bm tam + sbm ~ Consider three vectors, each with two elements denoted. Call the vectors ~a 1, ~a 2 and b. Call the elements of the first one a11 and a21, the elements of the second one a12 and a22 and the elements ~ of b, b1 and b2. Now consider two scalars denoted x1 and x2. Now multiply ~a 1 by x1 and ~a 2 by x2 and add the products. We obtain a11 a12 a11 x1 a12 x2 a11 x1 + a12 x2 x1 + x2 = + = (31) a21 a22 a21 x1 a22 x2 a21x1 + a22 x2 If set this expression equal to ~b we obtain a11 x1 + a12 x2 b1 = (32) a21x1 + a22 x2 b2 which is a linear system of 2 equations in 2 unknowns. We can write a general system of m equations in n unknowns as ~ x1 ~a 1 + x2 ~a 2 + ··· + xn ~a n = b (33) where xi are a series of scalar unknowns and each aj is a column of the A matrix of coefficients. 4.6. Addition of matrices. The sum C of a matrix A having m rows and n columns and a matrix B having m rows and n columns is a matrix having m rows and n columns whose elements are given by cij = aij + bij ∀ i, j (34) This gives 8 INTRODUCTION TO MATRIX ALGEBRA c11 c12 ... c1n a11 a12 ... a1n b11 b12 ... b1n c c ... c a a ... a b b ... b 21 22 2n 21 22 2n 21 22 2n .. .. .. C = = + (35) .. .. .. .. .. .. cm1 cm2 ... cmn am1 am2 ... amn bm1 bm2 ... bmn (36) a11 + b11 a12 + b12 ... a1n + b1n a + b a + b ... a + b 21 21 22 22 2n 2n .. = (37) .. .. am1 + bm1 am2 + bm2 ... amn + bmn 4.7. Inner (dot) product of two vectors. The inner (scalar or dot) product to two vectors u,v of length n is the scalar quantity denoted by n u · v = X uivi = u1v1 + u2v2 + ..
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