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# Example of an an idempotent matrix idempotent 1 0 0 0

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Unformatted text preview: rty: m×n n× p ′ AB, ( AB ) m×p p× m = ′ ′ ( B ) ( A ) p×n n×m p×m ′ ′ ≠ (A ) (B ) nm p× n × 29 Inverses Definition: Matrix A −1 is called inverse of a square matrix A if AA−1 = A−1 A = I Easy properties of inverses Existence: some square matrices do not have an inverse. some Such matrices are called singular. singular. Dimensionality: A and A −1 are of the same dimension. Uniqueness: An inverse matrix is unique if it exists. 30 More Inverse Properties The following three properties refer to nonsingular matrices A and B of The dimension n by n. dimension Inverse of an inverse: (A ) Inverse of a product: ( AB ) −1 = B −1 A−1 −1 − 1 Inverse of a transpose: ( A′) =A −1 =(A −1 ′ ) Try to write down the above properties in terms of the matrices’ Try elements and summation symbols to see how matrix notation is simplifying things. simplifying 31 Inverse Matrix and Linear Equation System 6 x1 +3 x2 +x3 =22 x1 +4 x2 −2 x3 =12 x −x +5 x =10 4 2 3 1 6 Matrix A = 1 Matrix 4 3 4 −1 6 x1 + 3 x2 + x3 22 x + 4 x − 2 x = d = 12 , Ax = ⇔ 1 2 3 10 4 x1 − x2 + 5 x3 Ax = d 1 − 2 is a square nonsingular matrix, so we can take an 5 inverse A−1 By definition of an inverse, By 1 1 A− Ax =A− d 1 x = A− d Since matrix A is nonsingular, its inverse A−1 exists and is unique, Since providing us with the unique vector of solution values. 32 Using Inverses to Solve Systems of Linear Equations We don’t know how to find inverse matrices (yet), so let us use an inverse We already computed for us: already 18 − 16 − 10 1 −1 A= − 13 26 13 52 − 17 18 21 Our solution then will be given by * x1 =2 18 − 16 − 10 22 2 * 1 12 = 3 −1 x=A d = 13 or x2 =3 − 13 26 52 x * =1 − 17 18 21 10 1 3 33 Finding Inverses Finding inverses can be tedious for matrices of large Finding dimensions dimensions Computationally finding an inverse can take literally Computationally weeks even with today’s computer technology weeks Numerous computational methods have been Numerous developed to reduce the time needed to compute inverses by taking into account specific structures of matrices in practical applications of Modern software allows one to invert matrices Modern quickly and easily for most practical uses quickly IIn the next chapter we shall learn how to find inverse n matrices, albeit not in the most efficient way matrices, 34 An Application: Markov Transition Matrices Definition: Markov transition matrix is a square matrix whose elements contain probability of transitions from one state to the other with the total number of states being equal to n, the Markov transition matrix’ dimension. PAA An example of a Markov transition matrix: M = PBA PAB PBB Let A and B be two industries. Assuming workers can move from one Let industry to another, PAB will be the probability of a worker currently working in industry A shifting to industry B next period. working PAA P AB PBA PBB = Probability of staying in A (moving from A to...
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