Lecture 30: Completing a Rank-One Matrix, Circulants

What this lecture covers

This lecture opens with a side question inspired by a course lab: given some prescribed entries of a matrix, when can the rest be filled in to make the whole matrix rank one? It answers this with a combinatorial rule using bipartite graphs, then switches topics to introduce circulant matrices and cyclic convolution, which set up the following lectures on eigenvectors of circulants and the Fourier matrix.

After watching, you can decide whether a partially filled matrix can be completed to rank one by checking for cycles in an associated bipartite graph, and you can explain what a circulant matrix is, why products of circulants are circulant, and how matrix multiplication of circulants corresponds to cyclic convolution of vectors.

Key ideas

• Rank-one matrix: any rank-one matrix has the form x*y^T (an outer product of two vectors), so M+N-1 entries are enough to determine all the rest, since one scaling degree of freedom is shared between the two vectors.
• Completion test via 2-by-2 determinants: for a rank-one matrix, every 2-by-2 submatrix must have determinant zero, so prescribed entries that already fill in a complete 2-by-2 block force what the remaining entries must be.
• Bipartite graph representation: representing each prescribed entry (i, j) as an edge between a row node and a column node turns the completion question into a graph problem.
• Completion rule: a set of prescribed positions can always be completed to a rank-one matrix (for generic non-zero values) if and only if its bipartite graph has no cycles; a cycle means the prescribed values over-determine the matrix and generically conflict.
• Circulant matrix: a matrix with constant diagonals that "wrap around" cyclically, determined entirely by its first column.
• Cyclic shift matrix: the basic circulant P shifts a vector's entries down by one position, with the last entry wrapping to the top; every circulant is a polynomial in P.
• Circulants form a closed family: because every circulant is a polynomial in P and P^n = I (the identity) for an n x n circulant, products of circulants are circulant, and multiplying them corresponds to cyclic convolution of their defining vectors, with powers of P beyond degree n-1 wrapping around.
• Shared eigenvectors: because every circulant is built from powers of the same shift matrix P, all circulants of a given size share the same eigenvectors, which turn out to be the eigenvectors of P.

Walkthrough

The rank-one completion question (0:00)

Prompted by a course lab on matrix completion, the lecture asks which sets of prescribed non-zero entries in a matrix can be filled in to make the whole matrix rank one. Using a 3-by-3 example with M+N-1 = 5 prescribed positions, it shows that a valid set of positions lets every remaining entry be recovered by forcing 2-by-2 determinants to zero, while an invalid set (one that already completes a 2-by-2 block) fails because it over-constrains a later entry.

Bipartite Graph (12:02)

The lecture reframes prescribed positions as edges in a bipartite graph connecting row nodes to column nodes. A worked 4-by-4 example shows that a set of positions whose graph contains a cycle cannot generally be completed to rank one, because the cycle encodes a redundant constraint that generic non-zero values will not satisfy.

Complete a Cycle with 6 Edges (20:20)

A second, less obvious failure case is constructed: a set of seven prescribed positions whose graph contains no completed 2-by-2 block (so no immediately visible contradiction) but does contain a longer cycle of six edges. This confirms the general rule, attributed to a combinatorics colleague, that completion is possible exactly when the bipartite graph of prescribed positions has no cycles. The lecture closes this topic by posing an open follow-up question about completing a matrix to rank two.

Convolutions and circulant matrices (24:19)

The lecture introduces circulant matrices: matrices with constant diagonals that cycle around, fully determined by their first column. It shows that the cyclic shift matrix P generates all circulants of a given size as polynomials c0*I + c1*P + c2*P^2 + ..., and that a product of two circulants is circulant because it is a product of polynomials in P, using the key identity P^n = I to keep the resulting polynomial's degree bounded.

Cyclic convolution of vectors (25:21)

Multiplying two polynomials that represent vectors corresponds to ordinary convolution, computed the way long multiplication works. The lecture works a numeric example, then shows how wrapping the highest-degree terms back around (using P^n = I) turns ordinary convolution into cyclic convolution, matching the result of multiplying the corresponding circulant matrices.

Eigenvectors of circulants (44:25)

Because every circulant is a polynomial in the shift matrix P, its eigenvectors are the same as those of P. The lecture identifies the all-ones vector as an eigenvector with eigenvalue 1, and the alternating (1, -1, 1, -1) vector as an eigenvector with eigenvalue -1 for a 4-by-4 example, noting that the remaining eigenvalues are complex fourth roots of unity, to be developed further in the next lecture.

Before you watch

• Be comfortable with rank-one matrices as outer products and with 2-by-2 determinants.
• Familiarity with basic graph terminology (nodes, edges, cycles) helps follow the bipartite graph argument.
• Some exposure to polynomial multiplication is useful for the convolution section.

Check your understanding

1. Why does a rank-one matrix need only M+N-1 prescribed entries to determine the rest?
2. Why does a completed 2-by-2 block among the prescribed entries make completion to rank one generically impossible?
3. What does a cycle in the bipartite graph of prescribed positions indicate about the completion problem?
4. Why is every circulant matrix a polynomial in the cyclic shift matrix P, and why does that make products of circulants also circulant?
5. How does multiplying two circulant matrices relate to cyclic convolution of their defining vectors?