Generalized Inverses and Solutions to Systems of Linear Equations
Throughout this book we will discover that in order to obtain “good” estimators of the elements of β (or functions thereof) under a linear model, and for other purposes as well, we need to solve various systems of linear equations of the general form $$\d
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Throughout this book we will discover that in order to obtain “good” estimators of the elements of β (or functions thereof) under a linear model, and for other purposes as well, we need to solve various systems of linear equations of the general form Ax = b. Here A is a specified n × m matrix called the coefficient matrix, b is a specified n-vector called the right-hand side vector, and x is an m-vector of unknowns. Any vector of unknowns that satisfies the system is called a solution. A solution may or may not exist. If n = m and A is nonsingular, solving the system of equations Ax = b is simple: A−1 b is the one and only solution, as is easily verified. But if A is not square, or if A is square but singular, how to find a solution, if indeed one (or more) exists, is not as obvious. In this chapter, we describe a general method for solving such systems. This method is based on the notion of a generalized inverse. The first section of the chapter gives the definition of a generalized inverse, and the second describes how a generalized inverse may be used to solve a system of linear equations. The third section presents a number of results that will be useful for developing the theory of linear models in subsequent chapters, and the last defines and gives some properties of a special generalized inverse.
© Springer Nature Switzerland AG 2020 D. L. Zimmerman, Linear Model Theory, https://doi.org/10.1007/978-3-030-52063-2_3
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3 Generalized Inverses and Solutions to Systems of Linear Equations
3.1
Generalized Inverses
Definition 3.1.1 A generalized inverse of an arbitrary matrix A is any matrix G for which AGA = A. Example 3.1-1. Generalized Inverses of Some Special Matrices T Consider the following matrices: In , 0n×m , u(n) i , 1n , and aa where a is any nonnull vector. Observe that
In In In = In , 0n×m 0m×n 0n×m = 0n×m , (n) (n)T (n)
(n)
(n)
ui ui ui = ui · 1 = ui , 1 1 T 1n 1n = 1n (1Tn 1n ) = 1n , 1n n n 1 1 aaT aaT = a(aT a)(aT a)aT = aaT , aaT a4 a4 1 T implying that In , 0m×n , u(n)T , n1 1Tn , and a 4 aa , respectively, are generalized i inverses of these matrices. Also observe that 11 10 11 11 = , 11 00 11 11
so
10 00
is a generalized inverse of J2 =
11 . 11
Does a generalized inverse exist for every matrix A? If A is nonsingular, then AA−1 A = A, so A−1 is a generalized inverse of A in this case. In fact, if A is nonsingular, then A−1 is its only generalized inverse, for if G1 and G2 are two generalized inverses of such an A, then AG1 A = AG2 A, implying (upon pre- and post-multiplying both sides by A−1 ) that G1 = G2 . If A is singular, however, then it can have more than one generalized inverse. For example, consider the matrix J2 . It can be verified that the following matrices are all generalized inverses of J2 :
10 , 00
01 , 00
0.5 0 , 0.5 0
0.5I2,
0.25J2.
(3.1)
(That the first matrix in this list is a generalized inverse of J2 was shown in Example 3.1-1.) Similarly, if A is nonsquare, it can have more than one generali
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