About Grand Magus
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I will start with my absolute favorite among all matrices. It has 2's
down the main diagonal and -1's on the diagonals just above and just
below. It is a Toeplitz matrix (constant diagonals), a second difference
matrix (because of -1, 2, -1), and a highpass filter. The matrix is
tridiagonal and positive definite, and you see it all over pure
mathematics too. Its determinant is n+1, and most important are its
eigenvectors which are pure sines. Recently, I came back to this
well-loved matrix, realizing that I didn't know its symmetric square
root, its exponential or its cosine. (Those are all needed for the
heat equation and wave equation.)
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