Stochastic matrix problem

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A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to $1$. Let $A$ be any (general) $2 times 2$ stochastic matrix.



a) Show that one of the eigenvalues of $A$ is $1$.



b) Show that a general $n times n$ stochastic matrix also has one eigenvalue equal to $1$.



c) What happens if the column sums are $1$ as above, but the entries may be any (possibly negative) real numbers?




I'm not sure how to start with this question. Should I explain using the Perron-Frobenius theorem? How to show for (a)? Because I think I can use the theorem only for (b), or is there a simpler explanation? And I'm not sure how to answer (c) as well.




matrices eigenvalues-eigenvectors stochastic-matrices




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    A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to $1$. Let $A$ be any (general) $2 times 2$ stochastic matrix.



    a) Show that one of the eigenvalues of $A$ is $1$.



    b) Show that a general $n times n$ stochastic matrix also has one eigenvalue equal to $1$.



    c) What happens if the column sums are $1$ as above, but the entries may be any (possibly negative) real numbers?




    I'm not sure how to start with this question. Should I explain using the Perron-Frobenius theorem? How to show for (a)? Because I think I can use the theorem only for (b), or is there a simpler explanation? And I'm not sure how to answer (c) as well.




    matrices eigenvalues-eigenvectors stochastic-matrices




    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite












      A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to $1$. Let $A$ be any (general) $2 times 2$ stochastic matrix.



      a) Show that one of the eigenvalues of $A$ is $1$.



      b) Show that a general $n times n$ stochastic matrix also has one eigenvalue equal to $1$.



      c) What happens if the column sums are $1$ as above, but the entries may be any (possibly negative) real numbers?




      I'm not sure how to start with this question. Should I explain using the Perron-Frobenius theorem? How to show for (a)? Because I think I can use the theorem only for (b), or is there a simpler explanation? And I'm not sure how to answer (c) as well.




      matrices eigenvalues-eigenvectors stochastic-matrices




      share|cite|improve this question















      A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to $1$. Let $A$ be any (general) $2 times 2$ stochastic matrix.



      a) Show that one of the eigenvalues of $A$ is $1$.



      b) Show that a general $n times n$ stochastic matrix also has one eigenvalue equal to $1$.



      c) What happens if the column sums are $1$ as above, but the entries may be any (possibly negative) real numbers?




      I'm not sure how to start with this question. Should I explain using the Perron-Frobenius theorem? How to show for (a)? Because I think I can use the theorem only for (b), or is there a simpler explanation? And I'm not sure how to answer (c) as well.





      matrices eigenvalues-eigenvectors stochastic-matrices





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      edited Aug 17 at 7:37









      Rodrigo de Azevedo

      12.6k41751




      12.6k41751










      asked Feb 20 '14 at 23:17









      user115636

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          Hint: a matrix $A$ has eigenvalue $1$ if and only if $det(A-I)=0$. Now look at $A-I$ and see what you get if you evaluate the determinant by adding the rows.






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            Hint: a matrix $A$ has eigenvalue $1$ if and only if $det(A-I)=0$. Now look at $A-I$ and see what you get if you evaluate the determinant by adding the rows.






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              Hint: a matrix $A$ has eigenvalue $1$ if and only if $det(A-I)=0$. Now look at $A-I$ and see what you get if you evaluate the determinant by adding the rows.






              share|cite|improve this answer






















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                Hint: a matrix $A$ has eigenvalue $1$ if and only if $det(A-I)=0$. Now look at $A-I$ and see what you get if you evaluate the determinant by adding the rows.






                share|cite|improve this answer












                Hint: a matrix $A$ has eigenvalue $1$ if and only if $det(A-I)=0$. Now look at $A-I$ and see what you get if you evaluate the determinant by adding the rows.







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                answered Feb 20 '14 at 23:21









                David

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