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We confirm a recent conjecture of Xin and Zhang [‘A variation of the Morris constant term’, Preprint, 2024, arXiv:2409.14356v2], which establishes a simple product formula for the characteristic polynomial of an $(n-1) \times (n-1)$ tridiagonal matrix. This characteristic polynomial arises from a recurrence relation that enumerates $n \times n$ nonnegative integer matrices with all row and column sums equal to t, also called the Ehrhart polynomial of the nth Birkhoff polytope.

As an extension of Sylvester’s matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form.

Summary

Chapter 7: Unitary matrices play important roles in theory and computation. The adjoint of a unitary matrix is its inverse, so unitary matrices are easy to invert. They preserve lengths and angles, and have remarkable stability properties in many numerical algorithms. In this chapter, we explore the properties of unitary matrices and present several special cases. We derive an explicit formula for a unitary matrix whose first column is given. We give a constructive proof of the QR factorization and show that every square complex matrix is unitarily similar to an upper Hessenberg matrix.

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A PROOF OF XIN–ZHANG’S TRIDIAGONAL DETERMINANT CONJECTURE

LEFT AND RIGHT EIGENVECTORS OF A VARIANT OF THE SYLVESTER–KAC MATRIX

7 - Unitary Matrices

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3 results

A PROOF OF XIN–ZHANG’S TRIDIAGONAL DETERMINANT CONJECTURE

LEFT AND RIGHT EIGENVECTORS OF A VARIANT OF THE SYLVESTER–KAC MATRIX

7 - Unitary Matrices

Summary