Intereting Posts

Showing $f_n(x):=\frac{x}{1+n^2x^2}$ uniformly convergent in $\mathbb R$ using $\epsilon-n_0$
Find polynomial $f(x)$ based on divisibility properties of $f(x)+1$ and $f(x) – 1$
Which prime numbers is this inequality true for?
Question About Orthoganality of Hermite Polynomials
What is the definition of a complex manifold with boundary?
Proof that operator is compact
Using Generating Functions to Solve Recursions
Categorical Pasting Lemma
Eigenvalues of a Permutation?
Isomorphism between complex numbers minus zero and unit circle
Second Countability of Euclidean Spaces
Is $\det(I+aAVV^*A^*)$ increasing function in $a$.
Approximation of $L^1$ function with compactly supported smooth function with same mass
Standard normal distribution hazard rate
Smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi direct product of cyclic groups.

It is well-known that every detailed-balance Markov chain has a diagonalizable transition matrix. I am looking for an example of a Markov chain whose transition matrix is not diagonalizable. That is:

Give a transition matrix $M$ such that there exists no invertible matrix $U$ with $U^{-1} M U$ a diagonal matrix.

Is there a combinatorial interpretation for the Jordan blocks that I can see directly from the graph?

- Complex Mean Value Theorem: Counterexamples
- Sum of closed spaces is not closed
- Mid-point convexity does not imply convexity
- Examples of Baire class 2 functions
- Counterintuitive examples in probability
- If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?

Edit: I found this example here: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-8-markov-eigenvalues-and-eigenvectors/MIT6_262S11_lec08.pdf on page 21.

\begin{array}{ccc}

1/2 & 1/2 & 0\\

0 & 1/2 & 1/2\\

0 & 0 & 1

\end{array}

This particular example has two non-recurrent states, which is not really what I want. So I am modifying my question to ask for an example of a Markov chain for which every state is recurrent.

- Where to start learning Linear Algebra?
- Should isometries be linear?
- Exponential bound on norm of matrix exponential (of linear ODE)
- Smash product of pointed spaces is not associative
- Does $\mathbb{R}^\mathbb{R}$ have a basis?
- What's the solution to $X = AXA$?
- Dimension of $\text{Hom}(U,V)$
- Eigenvalues for the rank one matrix $uv^T$
- $3\times 3$ matrix always has determinant $0$. Must $7$ of the elements be $0$?
- Formulas for the (top) coefficients of the characteristic polynomial of a matrix

Consider the matrix, $$M={1\over300}\pmatrix{210&40&24\cr15&210&96\cr75&50&180\cr}$$ Note that I adopt the convention that the columns, not the rows, are to add up to $1$. Now $1/2$ is an eigenvalue, since the first row of $$M-{1\over2}I={1\over300}\pmatrix{60&40&24\cr15&60&96\cr75&50&30\cr}$$ is four-fifths of the 3rd row. But also $1$ is an eigenvalue, and the eigenvalues add up to $2$, so $1/2$ is a repeated eigenvalue. Its eigenspace is one-dimensional, since $M-(1/2)I$ has rank $2$, so $M$ is not diagonalizable.

EDIT. I thought I’d have a go at finding an example with smaller numbers. Let $$M={1\over5}\pmatrix{2&2&1\cr1&2&1\cr2&1&3\cr}$$ The eigenvalues are $1$ (since the matrix is column-stochastic), $1/5$ (since the 1st and 3rd columns of $$M-{1\over5}I={1\over5}\pmatrix{1&2&1\cr1&1&1\cr2&1&2\cr}$$ are identical), and $1/5$ again (since the eigenvalues add up to $(2+2+3)/5=7/5$). $M-(1/5)I$ has rank $2$, so the eigenspace of the eigenvalue $1/5$ is 1-dimensional, so $M$ is not diagonalizable.

- Roots of $y=x^3+x^2-6x-7$
- Eigenvalue and Eigenvector for the change of Basis Matrix
- Find n that : $1+5u_nu_{n+1}=k^2, k \in N$
- Finding roots of $\sin(x)=\sin(ax)$ without resorting to complex analysis
- Help with proof using induction: $1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$
- In how many ways I can write a number $n$ as sum of $4$ numbers?
- Problem with the ring $R=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q &\Bbb Q\end{bmatrix}$ and its ideal $D=\begin{bmatrix}0&0\\ \Bbb Q & \Bbb Q\end{bmatrix}$
- Two Dimensional Lie Algebra
- Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$
- Let $p$, $q$ be prime numbers such that $n^{3pq} – n$ is a multiple of $3pq$ for all positive integers $n$. Find the least possible value of $p + q$.
- Probability/Combinatorics Question
- Quick question: tensor product and dual of vector space
- Subtracting a constant from log-concave function preserves log-concavity, if the difference is positive
- Deriving an expression for $\cos^4 x + \sin^4 x$
- Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?