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
add a comment |Â
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
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
edited Aug 17 at 7:37
Rodrigo de Azevedo
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asked Feb 20 '14 at 23:17
user115636
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8119
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1 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.
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
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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up vote
1
down vote
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.
add a comment |Â
up vote
1
down vote
up vote
1
down vote
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.
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.
answered Feb 20 '14 at 23:21
David
66.1k662125
66.1k662125
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