Demographic matrix models An eigenvalue eigenvector pair for the - - PowerPoint PPT Presentation

Demographic matrix models An eigenvalue ‐ eigenvector pair for the matrix A is any scalar  and vector w that satisfy ( h i ll i i h i ) Ei d   Aw w (technically w is a right eigenvector). Eigenvectors and eigenvalues are found by computer. Fact: A square matrix with k rows and k columns will possess k eigenvalue – eigenvector pairs. Fact: For most demographic projection matrices, there will be

• ne eigenvalue that is larger than all others. We call this

i l d it i t d ( i ht) i t th eigenvalue and its associated (right) eigenvector the “dominant” eigenvalue – eigenvector pair.

Demographic matrix models Fact: The dominant eigenvalue gives the long‐run finite rate of increase (), and the dominant (right) eigenvalue gives the stable age distribution stable age distribution. Fact: An eigenvalue – left eigenvector pair for the matrix A is l  d h i f any scalar  and vector v that satisfy

T T

  v A v Fact: The eigenvalues associated with the right eigenvectors are the same as the eigenvalues associated with the left eigenvector eigenvector. Fact: The left eigenvector associated with the dominant i l  i th d ti l eigenvalue  gives the reproductive values.

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photos courtesy Oregon State U