Taken from “Introduction to Linear Algebra” by Gilbert Strang:
I can verify this property of Markov matrices through computation on example matrices, but can someone please provide clarity on the final statement that 1 is an Eigenvalue?
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Hanzy
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4Hint: Find a vector such that its product with the matrix results in summing the matrix columns. – amd Jun 17 '18 at 01:51
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1The all-ones row vector is obviously a left eigenvector since each column sums to one, $1^TA=1^T$. Note that the eigenvalue associated with this eigenvector is one. And while the set of left and right eigenvectors are different, the set of eigenvalues is identical (since the characteristic polynomial is the same for $A^T$ as for $A$). – greg Jun 17 '18 at 03:34
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1Some related posts: Proof that the largest eigenvalue of a stochastic matrix is 111 and Why Markov matrices always have 1 as an eigenvalue. – Martin Sleziak Jun 17 '18 at 04:17
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1$\lambda$ is an eigenvalue if and only if $A-\lambda I$ is singular. Now substitute $\lambda=1$. – Jun 17 '18 at 06:07
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@Rahul Thabks for this reply. AlexanderJ93 pointed this out to me as well since I was looking at $A - I$ and failing to see the obvious coefficient of 1 in front of $I$. You both helped me greatly. – Hanzy Jun 17 '18 at 15:37
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Since each column of $A$ sums to $1$, each column of $A-I$ sums to $0.$ This means that the sum of the rows (linear combination with coefficients all equal to $1$), is the $0$ vector. If there is a linear combination of row vectors with not all zero coefficients, then the rows are linearly dependent, and any matrix with linearly dependent rows (or columns) must have determinant $0.$ Thus, $\det(A-I) = 0,$ so by definition, $\lambda_1 = 1$ is an eigenvalue.
Edit: Recall that $\lambda$ is an eigenvalue of $A$ if and only if $Av = \lambda v$ for some nonzero vector $v.$ Rearranging, we can see that this statement is equivalent to $(A-\lambda I)v = 0.$ If $A-\lambda I$ is invertible, then $v = (A-\lambda I)^{-1} \cdot 0 = 0,$ which is a contradiction. So, we must have that $A-\lambda I$ is not invertible, i.e. $\det(A-\lambda I) = 0.$ Thus, an alternate definition is $\lambda$ is an eigenvalue of $A$ if and only if $\det(A-\lambda I) = 0.$ Since we have that $\det(A-1\cdot I) = 0,$ $\lambda = 1$ is an eigenvalue.
Alex Jones
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I am still lost at the step “so by definition, lambda = 1 is an eigenvalue.” I’m sorry, I feel foolish as I’m certain I’m overlooking something that is in front of me (and already in your answer). But why is the eigenvalue 1? – Hanzy Jun 17 '18 at 02:19
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1I added some more information about it in the answer, please feel free to ask more if you still don't understand. – Alex Jones Jun 17 '18 at 02:34
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1thank you, I failed to see the (now obvious) coefficient of 1 in front of I in (A - I). Makes total sense now. – Hanzy Jun 17 '18 at 02:35
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Let $c(t)=det(tI_n-A)$ be the characteristic polynomial of the given matrix. Then we need to show that $c(1)=0$.
We have $$c(1)=det\pmatrix{1-a_{1,1}&.&.&-a_{1,n} \\\\\\-a_{1,n}&.&.&1-a_{1,n}}$$. Now do $R_1\to R_1+R_2+...+R_n$, the determinant remains unchanged and the top row is zero.