1,721,043 research outputs found
Doceamus: The Core Ideas in Our Teaching
No full textWhat will our students remember? One answer comes quickly but it is a counsel of despair: nothing at all. At the other extreme is an impossible hope that we all cherish: everything we say. Let me look for an intermediate answer, closer to reality, possibly by changing the question
Three steps on an open road
No full textThis note describes three recent factorizations of banded invertible infinite matrices: 1. If A has a banded inverse: A = BC with block--diagonal factors B and C. 2. Permutations factor into a shift times N<2w tridiagonal permutations. 3. A = LPU with lower triangular L, permutation P, upper triangular U. We include examples and references and outlines of proofs.National Science Foundation (U.S.) (Grant 1023152
Maximum flows and minimum cuts in the plane
Full text linkA continuous maximum flow problem finds the largest t such that div v = t F(x, y) is possible with a capacity constraint ||(v[subscript 1], v[subscript 2])|| ≤ c(x, y). The dual problem finds a minimum cut ∂ S which is filled to capacity by the flow through it. This model problem has found increasing application in medical imaging, and the theory continues to develop (along with new algorithms). Remaining difficulties include explicit streamlines for the maximum flow, and constraints that are analogous to a directed graph. Keywords: Maximum flow; Minimum cut; Capacity constraint; Cheege
The interplay of ranks of submatrices
Full text linkA banded invertible matrix T has a remarkable inverse. All "upper" and "lower" submatrices of T⁻¹ have low rank (depending on the bandwidth in T). The exact rank condition is known, and it allows fast multiplication by full matrices that arise in the boundary element method.
We look for the "right" proof of this property of T⁻¹. Ultimately it reduces to a fact that deserves to be better known: Complementary submatrices of any T and T⁻¹ have the same nullity. The last figure in the paper (when T is tridiagonal) shows two submatrices with the same nullity n – 3. Then C has rank 1. On and above the diagonal of T⁻¹, all rows are proportional.Singapore-MIT Alliance (SMA
Groups of banded matrices with banded inverses
Full text linkA product A=F[subscript 1]...F[subscript N] of invertible block-diagonal matrices will be banded with a banded
inverse. We establish this factorization with the number N controlled by the bandwidths w
and not by the matrix size n. When A is an orthogonal matrix, or a permutation, or banded
plus finite rank, the factors F[subscript i] have w=1 and generate that corresponding group. In the
case of infinite matrices, conjectures remain open
Fast transforms: Banded matrices with banded inverses
No full textIt is unusual for both A and A[superscript -1] to be banded—but this can be a valuable property in applications. Block-diagonal matrices F are the simplest examples; wavelet transforms are more subtle. We show that every example can be factored into A = F[subscript 1]…F[subscript N] where N is controlled by the bandwidths of A and A[superscript -1} (but not by their size, so this extends to infinite matrices and leads to new matrix groups)
Functions of Difference Matrices Are Toeplitz Plus Hankel
No full textWhen the heat equation and wave equation are approximated by and (discrete in space), the solution operators involve , , , and . We compute these four matrices and find accurate approximations with a variety of boundary conditions. The second difference matrix is Toeplitz (shift-invariant) for Dirichlet boundary conditions, but we show why also has a Hankel (anti-shift-invariant) part. Any symmetric choice of the four corner entries of leads to Toeplitz plus Hankel in all functions . Overall, this article is based on diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficients.National Science Foundation (U.S.) (Grant 1023152)MathWorks, Inc.Fulbright ProgramMIT Energy Initiative (Fellowship
Random Triangle Theory with Geometry and Applications
No full textWhat is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to (0,0) or reformulation as a 2 × 2 random matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage others to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the “square root ellipticity statistic” of random matrix theory. Another proof connects the Hopf map to the SVD of 2 × 2 matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight into both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed, and applications are proposed.National Science Foundation (U.S.) (NSF DMS 1035400)National Science Foundation (U.S.) (NSF DMS 1016125)National Science Foundation (U.S.) (NSF EFRI 1023152
The main diagonal of a permutation matrix
Full text linkBy counting 1’s in the “right half” of 2w consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth w. Then the matrix can be correctly centered and factored into block-diagonal permutation matrices.
Part II of the paper discusses the same questions for the much larger class of band-dominated matrices. The main diagonal is determined by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is determined “at infinity” in general, but from only 2w rows for banded permutations
The Jordan forms of AB and BA∗
Full text linkThe relationship between the Jordan forms of the matrix products AB and BA for
some given A and B was first described by Harley Flanders in 1951. Their non-zero eigenvalues and
non-singular Jordan structures are the same, but their singular Jordan block sizes can differ by 1.
We present an elementary proof that owes its simplicity to a novel use of the Weyr characteristic
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