MAT 2401 Linear Algebra

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MAT 2401Linear Algebra

• 3.4 Introduction to Eigenvalues

http//myhome.spu.edu/lauw
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HW

• Written Homework

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Overview

• Eigenvalues are used in a variety of real life
  applications.
• Eigenvalues are central to many theories for
  applicable mathematics.
• Eigenvalues is one of the most important topics
  in elementary linear algebra.
• More to come in section 7

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Notations

• Pay attention to the notations. They will be
  confusing numbers or vector?

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Example Preview Population Modeling (Leslie
Matrix)

• Suppose we are interested in the population of a
  certain type of bird in a forest area.
• We can divide the population in two age groups
  hatchlings (agelt1) and adults.

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Example Preview Population Modeling (Leslie
Matrix)

• Suppose we can estimate the following parameters
• Birth rate from hatchlings Bh
• Birth rate from adults Ba
• Survival rate of hatchlings Sh
• Survival rate of adults Sa

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Example Preview Population Modeling (Leslie
Matrix)

• We can model the population from year to year by
  the matrix equation

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Example Preview Population Modeling (Leslie
Matrix)

• Stable proportion of population in the age groups

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Example Preview Population Modeling (Leslie
Matrix)

• Q How to find ??
• A From the relationship Ax ?x.

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Eigenvalues and Eigenvectors

• Let A be a nxn matrix, ? a scalar, and x a
  non-zero nx1 column vector.
• ? and x are called an eigenvalue and eigenvector
  of A respectively if
• Ax ?x

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Example 1 (Eigenvector Given)

• Find the eigenvalue of A if the eigenvector is
• (a) (b)

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How to Find the Eigenvalues and Eigenvector?

• Recall Theorem from 3.3
• A is invertible if and only if det(A)?0
• Equivalently
• A is singular if and only if det(A)0
• Now,
• ?xAx

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Example 2

• Find the eigenvalues and eigenvectors of

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Remarks

• Eigenvalue and eigenvector always come in pairs.
• Eigenvectors are unique up to scalar multiple.

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Example 3

• Find the eigenvalues and eigenvectors of

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Remarks

• 1. det(?I-A)0 is called the _________________ of
  A.
• 2. It is a polynomial equation of degree ___.

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Visual Summary