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Chapter 2 a Short Review of Matrix Algebra

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Chapter 2 a Short Review of Matrix Algebra Chapter 2 A short review of matrix algebra 2.1 Vectors and vector spaces Definition 2.1.1. A vector a of dimension n is a collection of n elements typically written as a1 a2 a = = (a ) . . i n . an Vectors of length 2 (two-dimensional vectors) can be thought of points in the plane (See figures). A vector with all elements equal to zero is known as a zero vector and is denoted by 0. 31 BIOS 2083 Linear Models Abdus S. Wahed Figure 2.1: Vectors in two and three dimensional spaces (-1.5,2) (1, 1) (1, -2) x1 (2.5, 1.5, 0.95) x2 (0, 1.5, 0.95) x3 Chapter 2 32 BIOS 2083 Linear Models Abdus S. Wahed A vector whose elements are stacked vertically is known as column • vector whereas a vector whose elements are stacked horizontally will be referred to as row vector. (Unless otherwise mentioned, all vectors will be referred to as column vectors). A row vector representation of a column vector is known as its trans- • T pose. We will use the notation ‘′’ or ‘ ’ to indicate a transpose. For a1 2 a T instance, if a = and b = (a1 a2 ... a ), then we write b = a . n . an or a = bT . Vectors of same dimension are conformable to algebraic operations such • as additions and subtractions. Sum of two or more vectors of dimension n results in another n-dimensional vector with elements as the sum of the corresponding elements of summand vectors. That is, (ai)n (bi)n = (ai bi)n. ± ± Vectors can be multiplied by a scalar. • c(ai)n = (cai)n. Chapter 2 33 BIOS 2083 Linear Models Abdus S. Wahed Product of two vectors of same dimension can be formed when one of • them is a row vector and the other is a column vector. The result is called a1 b1 a2 b2 inner, dot or scalar product. if a = and b = , then . . . . an bn T a b = a1b1 + a2b2 + ... + anbn. Definition 2.1.2. The length, magnitude, or Euclidean norm of a vec- tor is defined as the square root of the sum of squares of its elements and is denoted by . For example, || || n 2 T a = (ai)n = a = √a a. || || || || v i u i=1 uX t The length of the sum of two or more vectors is less than or equal to the • sum of the lengths of each vector. (Cauchy-Schwarz Inequality). a + b a + b || || ≤ || || || || Chapter 2 34 BIOS 2083 Linear Models Abdus S. Wahed Definition 2.1.3. A set of vectors a1, a2,..., am is linearly dependent { } if at least one of them can be written as a linear combination of the others. In other words, a1, a2,..., am are linearly dependent if there exists at { } least one non-zero cj such that m cjaj = 0. (2.1.1) j=1 X In other words, for some k, ak = (1/ck) cjaj. − j=k X6 Definition 2.1.4. A set of vectors are linearly independent if they are not linearly dependent. That is, in order for (2.1.1) to hold, all cj’s must be equal to zero. Chapter 2 35 BIOS 2083 Linear Models Abdus S. Wahed Definition 2.1.5. Two vectors a and b are orthogonal if their scalar prod- uct is zero. That is, aT b = 0, and we write a b. ⊥ Definition 2.1.6. A set of vectors is said to be mutually orthogonal if members of any pair of vectors belonging to the set are orthogonal. If vectors are mutually orthogonal then they are linearly independent. • Chapter 2 36 BIOS 2083 Linear Models Abdus S. Wahed Definition 2.1.7. Vector space. A set of vectors which are closed under addition and scalar multiplication is known as a vector space. Thus if is a vector space, for any two vectors a and b from , (i) V V caa + cbb , and (ii) caa for any two constants ca and cb. ∈V ∈V Definition 2.1.8. Span. All possible linear combinations of a set of linearly independent vectors form a Span of that set. Thus if A = a1, a2,..., am is a set of m linearly independent vectors, { } then the span of A is given by m span(A) = a : a = cjaj , ( j=1 ) X for some numbers cj,j = 1, 2,...,m. Viewed differently, the set of vectors A generates the vector space span(A) and is referred to as a basis of span(A). Formally, Let a1, a2,..., am be a set of m linearly independent n-dimensional vec- • tor in a vector space that spans . Then a1, a2,..., am together forms V V a basis of and the dimension of a vector space is defined by the number V of vectors in its basis. That is, dim( ) = m. V Chapter 2 37 BIOS 2083 Linear Models Abdus S. Wahed 2.2 Matrix Definition 2.2.1. A matrix is a rectangular or square arrangement of num- bers. A matrix with m rows and n columns is referred to as an m n (read × as ‘m by n’) matrix. An m n matrix A with (i, j)th element aij is written × as a11 a12 ... a1n a21 a22 ... a2n A = (aij)m n = . × ... ··· ··· ··· am1 am2 ... amn If m = n then the matrix is a square matrix. Definition 2.2.2. A diagonal matrix is a square matrix with non-zero elements in the diagonal cells and zeros elsewhere. A diagonal matrix with diagonal elements a1, a2,...,an is written as a1 0 ... 0 0 a2 ... 0 diag(a1, a2,...,a ) = . n ... ··· ··· ··· 0 0 ... an Definition 2.2.3. An n n diagonal matrix with all diagonal elements equal × to 1 is known as identity matrix of order n and is denoted by In. Chapter 2 38 BIOS 2083 Linear Models Abdus S. Wahed A similar notation Jmn is sometimes used for an m n matrix with all × elements equal to 1, i.e., 1 1 ... 1 1 1 ... 1 J = = [1 1 ... 1 ] . mn m m m ... ··· ··· ··· 1 1 ... 1 Like vectors, matrices with the same dimensions can be added together and results in another matrix. Any matrix is conformable to multiplication by a scalar. If A = (aij)m n and B = (bij)m n, then × × 1. A B = (aij bij)m n, and ± ± × 2. cA = (caij)m n. × Definition 2.2.4. The transpose of a matrix A = (aij)m n is defined by × T A = (aji)n m. × If A = AT , then A is symmetric. • (A + B)T = (AT + BT ). • Chapter 2 39 BIOS 2083 Linear Models Abdus S. Wahed Definition 2.2.5. Matrix product. If A = (aij)m n and B = (aij)n p, × × then T AB = (cij)m p, cij = aikbkj = ai bj, × k X where ai is the ith row (imagine as a vector) of A and bj is the jth column (vector) of B. (AB)T = BT AT , • (AB)C = A(BC),whenever defined, • A(B + C) = AB + AC, whenever defined, • JmnJnp = nJmp. • Chapter 2 40 BIOS 2083 Linear Models Abdus S. Wahed 2.3 Rank, Column Space and Null Space Definition 2.3.1. The rank of a matrix A is the number of linearly inde- pendent rows or columns of A. We denote it by rank(A). rank(AT ) = rank(A). • An m n matrix A with with rank m (n) is said to have full row • × (column) rank. If A is a square matrix with n rows and rank(A) <n, then A is singular • and the inverse does not exist. rank(AB) min(rank(A),rank(B)). • ≤ rank(AT A) = rank(AAT ) = rank(A) = rank(AT ). • Chapter 2 41 BIOS 2083 Linear Models Abdus S. Wahed Definition 2.3.2. Inverse of a square matrix. If A is a square matrix with n rows and rank(A) = n, then A is called non-singular and there exists 1 1 1 1 a matrix A− such that AA− = A− A = In. The matrix A− is known as the inverse of A. 1 A− is unique. • If A and B are invertible and has the same dimension, then • 1 1 1 (AB)− = B− A− . 1 1 (cA)− = A− /c. • T 1 1 T (A )− = (A− ) . • Chapter 2 42 BIOS 2083 Linear Models Abdus S. Wahed Definition 2.3.3. Column space. The column space of a matrix A is the vector space generated by the columns of A. If A = (aij)m n = (a1 a2 ... an, × then the column space of A, denoted by (A) or (A) is given by C R n (A) = a : a = cjaj , C ( j=1 ) X for scalars cj,j = 1, 2,...,n. Alternatively, a (A) iff there exists a vector c such that ∈ C a = Ac. What is the dimension of the vectors in (A)? • C How many vectors will a basis of (A) have? • C dim( (A)) =? • C If A = BC, then (A) (B). • C ⊆ C If (A) (B), then there exist a matrix C such that A = BC. • C ⊆ C Example 2.3.1. Find a basis for the column space of the matrix 1 2 1 − − A = 1 1 4 . 0 2 2 Chapter 2 43 BIOS 2083 Linear Models Abdus S. Wahed Definition 2.3.4. Null Space. The null space of an m n matrix A is de- × fined as the vector space consisting of the solution of the system of equations Ax = 0. Null space of A is denoted by (A) and can be written as N (A) = x : Ax = 0 .
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