Math 335 Linear Algebra: Chapter 2 Determinants

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Math 335 Linear Algebra Chapter 2 Determinants

• Raul Cruz-Cano
• Texas AM-Texarkana/at NTCC
• Fall 2009

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Adjoint and Inverse Matrix

• A square matrix is invertible if and only if
  det(A)?0
• They can also be use to find the inverse of a
  matrix

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Eigenvectors and Eigenvalues (page 108)
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Eigenvectors and Eigenvalues
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Eigenvectors and Eigenvalues
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Section 11.6 Markov Chainspage 608

• Is a random process where all information about
  the future is contained in the present state
  (i.e. one does not need to examine the past to
  determine the future).
• In other words, future states depend only on the
  present state, and are independent of past
  states.
• The probability of moving from a present state j
  to a next state I is called transitional
  probability pi,j
• Transitional probabilities are often expressed in
  transition matrices in which element aj,i pi,j

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Example

• Example 1, page 609 A rental car agency has 3
  locations ..
• Notice that the probability of the car being
  return to station i depends only on the current
  station j, it does not matter in which stations
  was rented or returned in the past.

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Example

• Notice element a2,3 p3,2.6 means that the
  probability of going from location 3 to location
  2 is .6
• Notice that the sum of the elements in each
  column should always be equal to 1.

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State Vector for the nth observation

• A state vector is the vector of the probabilities
  that the system is in each state
• Examples
• Notice the sum of the elements in the vector is
  equal to 1
• Notice the dimension of the vector depends on the
  number of possible states.

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State Vector for the nth observation

• If P is the transition matrix of a Markov chain
  and x(n) is the state vector at the nth
  observation then x(n1) Px(n)
• Example

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Steady-State Vector

• The state vector q such the Pnxq as n?8 is
  called the steady-state
• What does it mean? Do you think that it is
  important to find it?
• To find q you need to multiply P by itself many
  many times or

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Steady-State Vector

• The steady state vector can be found by find a
  non-trivial solution for the system
• This equation should look familiar (think ?1,
  PA and qx)

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Examples

• Study Example 8 in page 616
• Solve Examples 1 and 2 in page 617