By Israel Gohberg, Peter Lancaster, Leiba Rodman

This booklet covers fresh ends up in linear algebra with indefinite internal product. It contains functions to differential and distinction equations with symmetries, matrix polynomials and Riccati equations. those functions are according to linear algebra in areas with indefinite internal product. The latter varieties an self sufficient department of linear algebra referred to as indefinite linear algebra. This new topic is gifted following the rules of a typical linear algebra course.

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**Example text**

13) If µ > 0 then i− (Ωn−1 ) = i− (Ωn−2 ) and since n+ (sn ) = n+ (sn−1 ) it follows that n+ (sn ) = i− (Ωn−1 ). Now, consider ﬁrst the case when det Ωn det Ωn−2 > 0. Then det Ωn , det Ωn−1 , det Ωn−2 have the same sign (positive or negative). 1 n+ (sn ) = n+ (sn−1 ) + 1. 3. One-Step Theorem 35 Taking into the consideration that i− (Ωn−1 ) = i− (Ωn−2 ) we obtain n+ (sn ) = n+ (sn−1 ) + 1 = i+ (Ωn−1 ) + 1 = i+ (Ωn ). Consider the second case when det Ωn det Ωn−2 < 0. Then det Ωn and det Ωn−1 have the same sign, and det Ωn−2 has the opposite sign.

This suggests that S may be chosen in such a way that both H and A are reduced to some simplest possible forms; an idea that will be developed in the next chapter. 48 Chapter 4. ) be an indeﬁnite inner product in Cn . 3)) if A = H −1 A∗ H. 6) Thus, any H-selfadjoint matrix A is similar to A∗ . , A = S −1 A∗ S for some S) then A is H-selfadjoint for some H. In other words, the similarity between A and A∗ can be carried out by means of an invertible hermitian matrix. , if A and B are H-selfadjoint then so is αA + βB where α, β are any real numbers.

A direct computation shows that A∗ QA = Q, so A is Q-unitary. Note also that K2 = K1−1 . 52 Chapter 4. Classes of Linear Transformations More generally, if K1 is an upper triangular Toeplitz matrix which is similar to one Jordan block with a nonzero eigenvalue, and if K2 = K1−1 , then 0 Sn K1 0 -unitary. 3 and therefore is omitted). 3. Let A be an H-unitary matrix. Then σ(A) is symmetric relative ∈ σ(A). , λ0 ∈ σ(A) implies λ−1 0 normal form of A, the sizes of Jordan blocks with eigenvalue λ0 , and the sizes of Jordan blocks with eigenvalue λ−1 0 , are the same.