Inverses and powers: Rules of Matrix Arithmetic

Transcription

1 Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix Theorem (Uniqueness of Inverse) Inverse Test Lemma (Reduced Row Echelon Form for Square Matrices) Lemma (Condition for Singularity) Lemma (AB=I and Nonsingularity) Proposition (AB=I implies BA=I) Theorem (A Right Inverse is an Inverse) New Inverse Test Theorem (Exponents and Transpose) Theorem (Inverse of Product of Matrices) 1.3 The Computation of the Inverse of a Matrix 1.4 Applying the Inverse of a Matrix to Systems of Linear Equations Theorem (Solving Equations Using the Matrix Inverse) 1.5 Powers of a Matrix Theorem (Laws of Exponents) Inverses and powers: Rules of Matrix Arithmetic What about division of matrices? We have considered addition, subtraction and multiplication of matrices. What about division? When we consider real numbers, we can write as In addition, we may think of as the multiplicative inverse of a, that is, it is the number which, when multiplied by a yields 1. In other words, if we set, then Finally, 1 is the multiplicative identity, that is, r1 = 1r = r for any real number r. While these concepts can not be extended to matrices completely, there are some circumstances when they do make sense. First, we can note that matrices satisfy [a] + [b] = [a + b] and [a][b] = [ab]. This means that both addition and multiplication of these matrices are just like the addition and multiplication of the real numbers. In this sense, matrices may be thought of as a generalization of the real numbers. Next we remember that if A is, then I m A = A = AI n. This means that the identity matrix (or, more properly, matrices) acts like 1 does for the real numbers. This also means that if we want there to be a (single) matrix I satisfying IA = A = AI, then we must have m = n. This means we have to restrict ourselves to square matrices. Inverses and powers: Rules of Matrix Arithmetic 1

2 If A is and matrix, then I n A = A = AI n, and so I n acts in the same manner as does 1 for the real numbers. Indeed, that is the reason it is called the identity matrix. Finally, we want to find (if possible) a matrix A 1 so that A 1 A = AA 1 = I. When such a matrix exists, it is called the inverse of A, and the matrix A itself is called invertible. Properties of the Inverse of a Matrix We consistently refer to the inverse of A rather than an inverse of A, which would seem to imply that a matrix can have only one inverse. This is indeed true. Theorem (Uniqueness of Inverse) A square matrix A can have no more than one inverse. Proof: Suppose we have matrices B and C which both act as inverses, that is, AB = BA = I and AC = CA = I. We evaluate BAC in two different ways and equate the results: BAC = (BA)C = IC = C and also BAC = B(AC) = BI = B, and so B = C. Inverse Test If A and B are square matrices of the same size, then B is a left inverse of A if BA = I. Similarly, it is a right inverse of A if AB = I. By definition A 1 is the inverse of A if AB = BA = I, that is, B is both a left inverse and a right inverse. We next make an observation about the reduced row echelon form of square matrices: Lemma (Reduced Row Echelon Form for Square Matrices) If A is an matrix then 1. The reduced row echelon form of A is I n, or 2. The last row of the reduced row echelon form of A is all zero. Proof: If every row in the reduced row echelon form of A has a leading one, then, since A has the same number of rows as columns, so does every column. This means that the leading ones must be on the diagonal, and the every other entry of the matrix is zero. In other words, the reduced row echelon form is I n. If, on the other hand, some row does not have a leading one, then it is an all-zero row. Since these rows are at the bottom of the matrix when it is in reduced row echelon form, the last row, in particular, must be all zero. When the reduced row echelon form is I n, the matrix is called nonsingular. Otherwise it is called singular. Next we give a criterion for nonsingularity. It is trivial that if then If this is the only vector x for which this is true, then M is nonsingular. What about division of matrices? 2

3 Lemma (Condition for Singularity) Math1300:MainPage/Inverses M is nonsingular if and only if implies Proof: First, suppose that M is nonsingular. The the equation reduced row echelon form, gives the equation. Hence has an augmented matrix which, in Now suppose that M is singular. The reduced row echelon form is not I n, and so some column does not contain a leading 1, that is, there must exist a free variable. It can be assigned a nonzero value, and thus provide a nonzero solution to Lemma (AB=I and Nonsingularity) If AB=I then B is nonsingular. Proof: Suppose that Multiply both sides of the equation by A to get On the other hand, A(Bx) = (AB)x = Ix = x, and so Hence implies and so B is nonsingular. Proposition (AB=I implies BA=I) Suppose the A and B are square matrices with AB = I. Then BA = I. Proof: From the previous lemma we know that B is nonsingular. Hence we know how to find C which is a solution to the equation BX = I, that is, so that BC = I. We now evaluate BABC in two different ways and equate the results: We get an important result from this Proposition. Theorem (A Right Inverse is an Inverse) Suppose A and B are square matrices with AB = I. Then B = A 1. Proof: By the Proposition above, AB = I implies BA = I. Since the inverse of A is unique, B = A 1. New Inverse Test If A and B are square matrices then B is the inverse of A if and only if AB = I. Here is an application of the previous theorem: Lemma (Condition for Singularity) 3

4 Theorem (Exponents and Transpose) If A is a square matrix with inverse A 1 then (A T ) 1 = (A 1 ) T Proof: Let B = (A 1 ) T. Then A T B = A T (A 1 ) T = (A 1 A) T = I T = I and so B = (A T ) 1. Here is another application of the previous theorem: Theorem (Inverse of Product of Matrices) If A and B are invertible matrices of the same size, then AB is also invertible and (AB) 1 = B 1 A 1 Proof: Since (AB)(B 1 A 1 ) = A(BB 1 )A 1 = AIA 1 = AA 1 = I, it follows that B 1 A 1 is the inverse of AB. The Computation of the Inverse of a Matrix Suppose we have a square matrix A and the reduced row echelon form of A is I (that is, A is nonsingular). X is the inverse of A if it satisfies the equation AX = I. We have seen how to solve such equations. We conclude that if we start with the matrix [A I] then the reduced row echelon form will be [I A 1 ]. This not only allows us to compute the inverse of A but it also shows that nonsingular matrices are invertible and vice-versa. Example: If we start with then has, as its reduced row echelon form, and so we conclude that Theorem (Exponents and Transpose) 4

5 Applying the Inverse of a Matrix to Systems of Linear Equations Theorem (Solving Equations Using the Matrix Inverse) If a system of linear equations is given by the equations Ax = b, and A has an inverse, then x = A 1 b. Proof: We take the equation Ax = b and multiply both sides by A 1 : A 1 (Ax) = A 1 b and A 1 (Ax) = (A 1 A)x = Ix = x, and so x = A 1 b. Example: Suppose we want to solve the system of equations Then let and so that we are solving Ax = b. We have already done the computation to determine that Hence and the (only) solution is x = 1,x = 1,x = 0. Powers of a Matrix Suppose we have a square matrix A. Then A 2 = AA, A 3 = AAA, and in general, for m,n > 0 we have 1 (Note that A = A) Applying the Inverse of a Matrix to Systems of Linear Equations 5

6 For invertible matrices, nonpositive powers are defined so that these equalities remain valid. n = 0: A m A 0 = A m + 0 = A m, so A 0 = I. m = 1,n = 1: A 1 A 1 = A = A 0 = I so A 1 is the inverse of A (hence the notation). n = m: A m A m = A m + ( m) = A 0 = I so A m is the inverse of A m. Theorem (Laws of Exponents) For any invertible square matrix A and integers m and n, A m A n = A m + n (A m ) n = A mn (A 1 ) 1 = A (A n ) 1 = (A 1 ) n Proof: The first two results have already been verified. The method for the last three is the same: (A 1 )A = I implies that A is the inverse of A 1, that is, A = (A 1 ) 1. Evaluate so the the middle factors "disappear": and so (A 1 ) n is the inverse of A n. so is the inverse of ra. Powers of a Matrix 6