INTRODUCTION TO MATRIX ALGEBRA. a 11 a a 1n a 21 a a 2n...

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1 INTRODUCTION TO MATRIX ALGEBRA 1 DEFINITION OF A MATRIX AND A VECTOR 11 Definition of a matrix A matrix is a rectangular array of numbers arranged into rows and columns It is written as a 11 a 12 a 1n a 21 a 22 a 2n (1) a m1 a m2 a mn 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 a ij (2) 12 Definition of a vector A vector is a n-tuple of numbers In R 2 a vector would be an ordered pair of numbers {x, y} InR 3 a vector is a 3-tuple, ie, {x 1,x 2,x 3 } Similarly for R n Vectors are usually denoted by lower case letters such as a or b, or more formally a or b 13 Row and column vectors 131 Row vector A matrix with one row and n columns (1xn) is called a row vector It is usually written x or x x 1,x 2,x 3,, x n (3) The use of the prime 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 132 Column vector A matrix with one column and n rows (nx1) is called a column vector It is written as x 1 x 2 x 3 x (4) Note that each column of a matrix is a column vector It is common to write the columns of a matrix as a 1,a 2, a n where each column vector a j is of length m As an example a 2 is given by x n Date: August 27,

2 2 INTRODUCTION TO MATRIX ALGEBRA a 2 a 12 a 22 a 32 a m2 (5) 2 VARIOUS TYPES OF MATRICES AND VECTORS 21 Square matrices A square matrix is a matrix with an equal number of rows and columns, ie mn 22 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 A or A T and If a ij is the ij th element of A, then a ij then A is given by A a ji when A a ij (6) A A a ji If the matrix A is given by Symmetric matrix A symmetric matrix is a square matrix A for which (7) (8) An example of a symmetric matrix is T T A A (9) (10)

3 INTRODUCTION TO MATRIX ALGEBRA 3 24 Identity matrix The identity matrix of order n written I or I n, is a square matrix having ones along the main diagonal (the diagonal running from upper left to lower right and zeroes elsewhere) If we write I δ ij then δ ij { 1, i j 0, i j (11) (12) The symbol δ ij 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 25 Scalar matrix For any scalar λ, the square matrix S λδ ij λi (13) is called a scalar matrix An example is (14) Diagonal matrix A square matrix D λ i δ ij (15) is called a diagonal matrix Notice that λ i varies with i An example is (16) If a system of equations was written with this coefficient matrix, we could solve the system by solving each equation individually 27 Null or zero matrix The null or zero matrix is a matrix with each element being zero It is denoted as 0

4 4 INTRODUCTION TO MATRIX ALGEBRA Upper triangular matrix A matrix with all elements below the main diagonal equal to zero is called an upper triangular matrix a 11 a 12 a 13 a 1n 0 a 22 a 23 a 2n 0 0 a 33 a 3n A (18) a mn Specifically a ij 0if i>jas long as i<mand j<n 29 Lower triangular matrix A matrix with all elements above the main diagonal equal to zero is called a lower triangular matrix a a 21 a a 31 a 32 a 33 0 A (19) a m1 a m2 a m3 a mn Specifically a ij 0if i<jas long as i<mand j<n 3 A NOTE ON SUMMATION NOTATION 31 Single sums 311 Definition of a single sum a i a m + a m+1 + a m a n (20) 312 Properties of a single sum im ka i k (17) a i (21) k k + k + k + + k nk (a i + b i ) a i + b i

5 INTRODUCTION TO MATRIX ALGEBRA 5 32 Double sums 321 Definition of a double sum m j1 a ij m j1 a 1j + m j1 a 2j + + m j1 a nj (22) 322 Properties of a double sum ( a j )( a i ) j1 a 11 + a 12 + a a 1m +a 21 + a 22 + a a 2m a n1 + a n2 + a n3 + + a nm a 2 i a 2 i +2 a i a j (23) i<j + a i a j i j

6 6 INTRODUCTION TO MATRIX ALGEBRA 4 MATRIX OPERATIONS 41 Scalar multiplication (matrix) Given a matrix A and a scalar λ, the product of λ and A, written λa, is defined to be λa 11 λa 12 λa 1n λa 21 λa 22 λa 2n λa (24) λa m1 λa m2 λa mn 42 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 λ a λa 1 λa 2 λa m For the second column of a matrix we could write λ a 2 λa 12 λa 22 λa 32 λa m2 (25) (26) 43 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 C (27) The trace of C is [ ] 3 44 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 This gives c j a j + b j j (28)

7 INTRODUCTION TO MATRIX ALGEBRA 7 c c 1 c 2 c m a 1 a 2 a m + b 1 b 2 b m a 1 + b 1 a 2 + b 2 a m + b m (29) 45 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, t a 1 a 2 a m + s b 1 b 2 b m ta 1 + sb 1 ta 2 + sb 2 ta m + sb m (30) Consider three vectors, each with two elements denoted Call the vectors a 1, a 2 and b Call the elements of the first one a 11 and a 21, the elements of the second one a 12 and a 22 and the elements of b, b 1 and b 2 Now consider two scalars denoted x 1 and x 2 Now multiply a 1 by x 1 and a 2 by x 2 and add the products We obtain a11 x 1 a 21 a12 + x 2 a 22 a11 x 1 a 21 x 1 + a12 x 2 a 22 x 2 a11 x 1 + a 12 x 2 a 21 x 1 + a 22 x 2 (31) If set this expression equal to b we obtain a11 x 1 + a 12 x 2 a 21 x 1 + a 22 x 2 ( b1 b 2 ) (32) which is a linear system of 2 equations in 2 unknowns We can write a general system of m equations in n unknowns as x 1 a 1 + x 2 a x n a n b (33) where x i are a series of scalar unknowns and each a j is a column of the A matrix of coefficients 46 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 This gives c ij a ij + b ij i, j (34)

8 8 INTRODUCTION TO MATRIX ALGEBRA c 11 c 12 c 1n a 11 a 12 a 1n b 11 b 12 b 1n c 21 c 22 c 2n a 21 a 22 a 2n b 21 b 22 b 2n C + c m1 c m2 c mn a m1 a m2 a mn b m1 b m2 b mn a 11 + b 11 a 12 + b 12 a 1n + b 1n a 21 + b 21 a 22 + b 22 a 2n + b 2n a m1 + b m1 a m2 + b m2 a mn + b mn (35) (36) (37) 47 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 u v u i v i u 1 v 1 + u 2 v u n v n (38) 48 Multiplication of matrices Given an mxn matrix A and an nxr matrix B, the product AB is defined to be an mxr matrix C, whose elements are computed from the elements of A,B according to c ij a ik b kj,,, m, j 1,, r (39) k1 In other words to obtain the ijth element of c we take the ith row of A and jth column of B and form the inner product As an example consider the matrices below A B (40) The element c 11 comes from multiplying the first row of A with the first column of B as follows: c 11 ( ) (41) 1 Similarly the element c 32 comes from multiplying the third row of A with the second column of B as follows: c 32 ( ) (42) 4 Multiplying out the rest of the entries gives

9 INTRODUCTION TO MATRIX ALGEBRA 9 C (43) Some properties of matrix operations Let α and β denote real numbers (scalars), a, b, c denote n-vectors, and A, B, C denote matrices The properties are conditional on the operations being defined for the case in point 491 Equality vectors: Two n-vectors a and b are said to be equal if all their corresponding components are equal Equality is only possible for vectors of the same dimension matrices: Two m x n matrices A and B are said to be equal if all their corresponding components are equal Equality is only possible for matrices of the same dimension 492 Multiplication by a scalar a: ( α + β )AαA+βA b: α(a + B) αa +αb c: α (βa) (αβ) A Note that A and B above can be replaced by a and b as in (1)( a) a 493 Addition a: a + b b + a b: a +0 a c: ( a + b)+ c a +( b + c ) d: a +( a) 0 e: A + B B + A f: A +(B + C) (A + B)+C g: A +00+A A h: A +( A) Multiplication a: a b b a b: AB BA c: A(BC) (AB)C d: α( b + c )α b + α c e: A(B + C) AB + AC f: (B + C)A BA + CA g: (α a) b a (α b)α( a b) h: a a > 0 a 0 i: a 00 a 0 j: A0 0A 0 k: AI IA A 495 Transposes a: (A ) A b: (ABC) C B A c: (A + B) A + B

10 10 INTRODUCTION TO MATRIX ALGEBRA 496 Properties of the trace a: trace (I) n b: trace (ABC) trace (CAB) trace (BCA) c: trace (A + B) trace (A) + trace (B) d: tr(ab) tr(ba) if both AB and BA are defined e: tr(ka) ktr(a) where k is a scalar 410 Idempotent matrices - A matrix is called idempotent if A 2 A (44) For example the identity matrix is idempotent Consider the matrix M below M (45) We can verify that it is idempotent by carrying out the multiplication MM Consider the multiplication of the first row and first column (47) Or consider the multiplication of the first row and second column (48) Note that if A is idempotent, tr(a) rank of A (46)