This solution offers additional insights into the theory of block-tridiagonal Toeplitz matrices. Block Toeplitz matrices have constant blocks on each block diagonal parallel to the block main diagonal. A block partitioned matrix is said to be block-tridiagonal if the nonzero blocks occur only on the block subdiagonal, the block main diagonal, and the block superdiagonal. Block-tridiagonal Toeplitz matrices are particularly nice in that they are inexpensive to investigate. Our first observation on such particular block Toeplitz matrices is easy to check, and its proof is therefore left to the reader.

THEOREM 1. For any given pair of real m × k matrices A and B, consider the following block partitioned matrices:

where M consists of n × n blocks, whereas N has n × (n + 1) blocks. Then

thus particularly showing that the symmetric block-tridiagonal Toeplitz matrix M is a nonnegative definite matrix satisfying

, with

denoting the range (column space), the null space, and the transpose, respectively, of the matrix (·).

By assuming a disjointness property between the matrices A and B, Theorem 1 can even be strengthened as follows.

THEOREM 2. Let A and B be real matrices of the same order satisfying

. Let M and N be defined in terms of A and B as in Theorem 1. Then

where

Proof. Theorem 1 tells us that

. Now, let

be partitioned in accordance with N^t. Then B^tx₁ = 0, A^tx_n = 0, and, for i = 1,…,n − 1, A^tx_i + B^tx_i+1 = 0. Clearly, if

or if

, then this set of n + 1 equations is satisfied if and only

for i = 1,2,…,n. Hence, because of

or, equivalently, as claimed

; recall that M and W are symmetric matrices and observe that

holds for any matrix G, where (·)^⊥ indicates the orthogonal complement of (·) with respect to the usual inner product. █

According to Werner (1986), two real m × k matrices, say, A and B, are said to be weakly complementary to each other if

. Such matrices are of great importance, for instance, in statistics (cf. Werner, 1985b, 1987). If A and B are such a pair of weakly complementary matrices, then trivially

, and so, by virtue of Theorem 2,

. Next, let A and B be symmetric matrices. Moreover, let at least one of these matrices, say, A, be nonnegative definite. Then it is well known that

(cf. Werner, 1985a, Lemma 3.2; or the theorem in Werner, 2003). Hence, because of Theorem 2, again

. Finally, as in Tian's posed problem, let A and B be orthogonal projectors. Note that a real matrix P is an orthogonal projector if and only if P = P^t and P² = P, that is, if and only if P is symmetric and idempotent. Then P = P² = PP^t, thus showing that P is also nonnegative definite. Consequently, by virtue of the previous case, again

. In other words, Tian's claim follows as a very special case from our more general result stated in Theorem 2.

Throughout this text we considered only real matrices. Needless to say, all results can be generalized to complex matrices just by replacing symmetric matrices and transposition by Hermitian matrices and conjugate transposition, respectively.

We conclude with mentioning that Theorem 2 also generalizes IMAGE Problem 31-3 by Tian (2003).