Unit 11: Complex Vectors and Markov Matrices
What changes when a vector's entries turn complex, and how a matrix whose columns sum to 1 predicts where a population settles.
6 topics · 22 questions
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Complex Vectors and the Hermitian Dot Product
- How it worksHow does conjugating one vector before dotting it with another keep the length of a complex vector from turning negative?
- Worked exampleFind the Hermitian dot product of the complex vectors (1 + i, 2) and (1 - i, 3).
- Common mistakeWhy doesn't the ordinary dot product formula give a sensible notion of length for a complex vector?
- Going furtherWhat happens to the Hermitian dot product of a complex vector with itself?
Your turn1 of 2
Find the Hermitian dot product of the complex vector (3 + 2i, 7 + -4i) with itself.Complex Numbers as 2x2 Rotation Matrices
- How it worksHow does representing a complex number as a 2 by 2 matrix turn complex multiplication into ordinary matrix multiplication?
- Worked exampleFind the 2 by 2 matrix representing the complex number 3 + 4i and use it to rotate and scale the vector (1, 0).
- Ties togetherHow does this matrix picture of complex numbers connect back to the rotation matrices seen earlier in the course?
Your turn1 of 2
Find the magnitude of the complex number 4 + 3i, which is also the scaling factor of its matrix representation.Complex Eigenvalues of Real Matrices
- How it worksHow does a real characteristic polynomial force its complex roots to come in conjugate pairs?
- Worked exampleFind the complex eigenvalues of the matrix [[1, -1], [1, 1]].
- Ties togetherHow do these complex conjugate eigenvalue pairs connect back to the rotation matrices seen earlier?
Your turn1 of 2
Find the sum of the two complex eigenvalues of the matrix [[4, -6], [6, 4]].Markov Matrices and Transition Probabilities
- How it worksHow does multiplying a probability vector by a Markov matrix redistribute it among the possible states one step later?
- Worked exampleFind the state after one step given the Markov matrix [[0.9, 0.2], [0.1, 0.8]] and the starting vector (1, 0).
- Common mistakeWhy doesn't a Markov matrix need every entry to be equal for its probabilities to make sense?
- Going furtherWhat happens to a probability vector after two steps instead of just one under a Markov matrix?
Your turn1 of 2
A Markov matrix has columns (0.657, 1-0.657) and (0.619, 1-0.619). Find the first entry of the state after one step, starting from (1, 0).Steady-State Vectors of a Markov Chain
- How it worksHow does the eigenvalue 1 of a Markov matrix always come with an eigenvector that's the chain's steady state?
- Worked exampleFind the steady-state vector of the Markov matrix [[0.9, 0.2], [0.1, 0.8]].
- Worked exampleShow that (4/7, 3/7) is the steady-state vector of the Markov matrix [[0.7, 0.4], [0.3, 0.6]].
- Common mistakeWhy doesn't a Markov chain's steady state depend on which state it started in?
Your turn1 of 2
A Markov matrix has columns (0.657, 1-0.657) and (0.619, 1-0.619). Find the first entry of its steady-state vector, normalized to sum to 1.Long-Run Behavior of Repeated Markov Steps
- How it worksHow does writing the starting vector in terms of eigenvectors show every eigenvalue except 1 dying out over time?
- Worked exampleFind the approximate state after many steps of the Markov matrix [[0.9, 0.2], [0.1, 0.8]] starting from (0, 1).
- Common mistakeWhy doesn't every Markov chain reach the same steady state no matter which state it starts in?
- Going furtherWhat happens to the long-run behavior when a Markov matrix has a second eigenvalue equal to negative 1?
Your turn1 of 2
A Markov matrix has columns (0.657, 1-0.657) and (0.619, 1-0.619). After many steps, the state converges to the steady-state vector regardless of the start. Find the second entry of that steady-state vector.