Quadratic Mathematics | Part IB, 2001

Let J1J_{1} denote the 2×22 \times 2 matrix (0110)\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right). Suppose that TT is a 2×22 \times 2 uppertriangular real matrix with strictly positive diagonal entries and that J11TJ1T1J_{1}^{-1} T J_{1} T^{-1} is orthogonal. Verify that J1T=TJ1J_{1} T=T J_{1}.

Prove that any real invertible matrix AA has a decomposition A=BCA=B C, where BB is an orthogonal matrix and CC is an upper-triangular matrix with strictly positive diagonal entries.

Let AA now denote a 2n×2n2 n \times 2 n real matrix, and A=BCA=B C be the decomposition of the previous paragraph. Let KK denote the 2n×2n2 n \times 2 n matrix with nn copies of J1J_{1} on the diagonal, and zeros elsewhere, and suppose that KA=AKK A=A K. Prove that K1CKC1K^{-1} C K C^{-1} is orthogonal. From this, deduce that the entries of K1CKC1K^{-1} C K C^{-1} are zero, apart from nn orthogonal 2×22 \times 2 blocks E1,,EnE_{1}, \ldots, E_{n} along the diagonal. Show that each EiE_{i} has the form J11CiJ1Ci1J_{1}{ }^{-1} C_{i} J_{1} C_{i}^{-1}, for some 2×22 \times 2 upper-triangular matrix CiC_{i} with strictly positive diagonal entries. Deduce that KC=CKK C=C K and KB=BKK B=B K.

[Hint: The invertible 2n×2n2 n \times 2 n matrices SS with 2×22 \times 2 blocks S1,,SnS_{1}, \ldots, S_{n} along the diagonal, but with all other entries below the diagonal zero, form a group under matrix multiplication.]

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