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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 Fall '11
 Kim
 Linear Algebra, Matrices, Inverses

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