Eigenvalues
A 3×3 matrix A has eigenvalues 2, 2, and 5. What is the determinant of A?
The determinant of A is the product of its eigenvalues, so det(A)=2×2×5=20
What is a stochastic matrix, and what must each column (or row) of the matrix sum to?
A stochastic matrix represents transition probabilities between states. Each column (or row, depending on the setup) must sum to 1, since probabilities must total 1.
What geometric object is represented by the vector equation r=a+tb ?
It represents a line that passes through the point a and is parallel to the direction vector b.
Explain why every upper-triangular matrix has its eigenvalues on the main diagonal.
Because for a triangular matrix, det(A−λI) equals the product of the diagonal entries minus λ, so the roots of the characteristic polynomial are exactly those diagonal entries
Why does every Markov chain have an eigenvalue equal to 1?
Because probabilities are preserved under transitions, so the steady-state vector satisfies Ap=p, meaning 1 is always an eigenvalue.
The plane through a point 'a' with normal vector n can be written as n⋅(r−a)=0 Explain what this equation means geometrically.
It means all position vectors r that satisfy this equation make the same angle with n— they lie in a plane perpendicular to n and passing through a
If A is diagonalizable, what does that tell us about its eigenvectors, and how does diagonalization simplify A^n?
Diagonalizability means A has a full set of linearly independent eigenvectors, forming a basis of R^n. Then A=PDP^(−1), and A^n=P(D^n)P^(−1), making powers of A easy to compute.
What does the steady-state vector of a Markov chain represent, and how is it found?
It represents the long-run proportion of time spent in each state. It’s found by solving Ap=pwith the condition that the components of p sum to 1.
Explain how to determine whether a line and a plane intersect, are parallel, or lie entirely in the same plane.
Substitute the line equation r=a+tbb into the plane equation n⋅(r−r0)=0
• If there is one solution for t, the line intersects the plane.
• If there is no solution and n⋅b=0, the line is parallel to the plane.
• If all t satisfy the equation, the line lies in the plane.
A 3×3 real matrix has one real eigenvalue and two complex conjugate eigenvalues. What can you say about its trace and determinant?
The trace equals the sum of all three eigenvalues (so it’s real), and the determinant equals their product (also real). Complex eigenvalues always appear in conjugate pairs for real matrices.
A transition matrix T has eigenvalues 1,0.2,0.1.
Explain why the population distribution x_n=T^nx_0 always tends toward the same limiting vector, regardless of the initial state x_0.
Since 1 is the largest eigenvalue and all others 0.2, 0.1 have absolute value less than 1, the terms involving those eigenvalues decay to zero as n→∞
Two planes have equations n1⋅r=d1 and n2⋅r=d2
Explain how you can determine whether they are parallel, identical, or intersecting.
Compare their normal vectors:
• If n1 is a multiple of n2 and d1≠kd, the planes are parallel but distinct.
• If n1 is a multiple of n2 and d1=kd, they are the same plane.
• If n1 and n2 are not multiples, they intersect in a line.
A symmetric matrix A has two distinct eigenvalues. What can you say about the relationship between their eigenvectors?
For a real symmetric matrix, eigenvectors corresponding to different eigenvalues are orthogonal.
In a Markov chain represented by transition matrix T, explain how the eigenvector corresponding to eigenvalue 1 determines the system’s long-term behaviour.
Then describe what would happen if the entries of that eigenvector were not all positive.
The eigenvector corresponding to eigenvalue 1 gives the steady-state distribution of the Markov chain — it shows the proportion of time the system spends in each state once equilibrium is reached.
If some entries of that eigenvector were not positive, it would mean the vector cannot represent valid probabilities, since probabilities must be non-negative and sum to 1. In that case, the system would not have a valid steady-state distribution
Three planes in ℝ³ can intersect in several ways. (State the rank) and Explain the geometric conditions under which they:
(a) intersect at a single point,
(b) intersect along a line, or
(c) have no common intersection.
(a) Single point: their normal vectors are not coplanar (the matrix of normals has rank 3).
(b) Line: normals are coplanar but not all parallel (rank 2).
(c) No common intersection: two or more planes are parallel or inconsistent (rank < 2 or inconsistent right-hand sides).