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This chapter deals with nonnegative matrices, which are relevant in the study of Markov processes because the state transition matrix of such a process is a special kind of nonnegative matrix, known as a stochastic matrix. However, it turns out that practically all of the useful properties of a stochastic matrix also hold for the more general class of nonnegative matrices. Hence it is desirable to present the theory in the more general setting, and then specialize to Markov processes. The chapter first considers the canonical form for nonnegative matrices, including irreducible matrices and periodic irreducible matrices, before discussing the Perron–Frobenius theorem for primitive matrices and for irreducible matrices.

Group actions together with properly formulated objective functions can offer a channel to tackle various classical or new and challenging problems rising from applied linear algebra. This chapter presents a framework to bring together the notions of group theory, linear transformations, and dynamical systems as a tool to undertake the task of system identification by canonical forms.

This chapter studies matrices (or linear transformations) that are selfadjoint or skewadjoint with respect to a nondegenerate hermitian or skewhermitian inner product. As an application of the canonical forms obtained in chapters 8 and 9, canonical forms for such matrices are derived in this chapter. Matrices that are skewadjoint with respect to skewhermitian inner products are known as Hamiltonian matrices; they play a key role in many applications such as linear control systems. The canonical forms reveal invariant Lagrangian subspaces; in particular, they give criteria for existence of such subspaces. Another application involves boundedness and stable boundedness of linear systems of differential equations with constant coefficients under suitable symmetry requirements.

This chapter also studies the canonical forms of mixed quaternion matrix pencils, i.e., such that one of the two matrices is φ‎-hermitian and the other is φ‎-skewhermitian, with respect to simultaneous φ‎-congruence. It starts by formulating the canonical form for φ‎-hsk matrix pencils under strict equivalence. Other canonical forms of mixed matrix pencils are developed with respect to strict equivalence. As an application, this chapter provides canonical forms of quaternion matrices under φ‎-congruence. As in the preceding chapter, this chapter also fixes a nonstandard involution φ‎ throughout and a quaternion β‎(φ‎) such that φ‎=(β‎(φ‎)) = −β‎(φ‎) and ∣β‎(φ‎)∣ = 1.

This chapter turns to matrix pencils of the form A + tB, where one of the matrices A or B is skewhermitian and the other may be hermitian or skewhermitian. Canonical forms of such matrix pencils are given under strict equivalence and under simultaneous congruence, with full detailed proofs, again based on the Kronecker forms. Comparisons with real and complex matrix pencils are presented. In contrast to hermitian matrix pencils, two complex skewhermitian matrix pencils that are simultaneously congruent under quaternions need not be simultaneously congruent under the complex field, although an analogous property is valid for pencils of real skewsymmetric matrices. Similar results hold for real or complex matrix pencils A + tB, where A is real symmetric or complex hermitian and B is real skewsymmetric or complex skewhermitian.

This chapter links Krylov subspace methods to classical topics of linear algebra. The main goal is to explain when a Krylov sequence can be orthogonalised with an optimal (Arnoldi-type) short recurrence. This question was posed by Golub in 1981 and answered by the Faber–Manteuffel theorem in 1984. The chapter gives a new complete proof of this theorem that has not been published elsewhere. It is based on the cyclic decomposition of a vector space with respect to a given linear operator. The theorem motivates the theoretically and practically important distinction made between Hermitian and non-Hermitian problems in the area of Krylov subspace methods. The matrix-version of the theorem works with the so-called B-normal(s) matrices, and this property is linked for a general matrix with the number of its distinct eigenvalues. The chapter also reviews other types of recurrences and it ends with brief remarks on integral representations of invariant subspaces.

This chapter fixes a nonstandard involution φ‎. It introduces indefinite inner products defined on H^n×1 of the symmetric and skewsymmetric types associated with φ‎ and matrices having symmetry properties with respect to one of these indefinite inner products. The development in this chapter is often parallel to that of Chapter 10, but here the indefinite inner products are with respect to a nonstandard involution, rather with respect to the conjugation as in Chapter 10. This chapter develops canonical forms for (H,φ‎)-symmetric and (H,φ‎)-kewsymmetric matrices (when the inner product is of the symmetric-type), and canonical forms of (H,φ‎)-Hamiltonian and (H,φ‎)-skew-Hamiltonian matrices (when the inner product is of the skewsymmetric-type). Applications include invariant Lagrangian subspaces and systems of differential equations with symmetries.

This chapter presents canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are well-known. The chapter first looks at Jordan and Kronecker canonical forms, before turning to real matrix pencils with symmetries. It provides canonical forms for pairs of real matrices, either one of which is symmetric or skewsymmetric, or what is the same, corresponding matrix pencils. Finally, this chapter presents canonical forms of complex matrix pencils with various symmetries, such as complex matrix pencils with symmetries with respect to transposition.

This chapter treats matrix polynomials with quaternion coefficients. A diagonal form (known as the Smith form), which asserts that every quaternion matrix polynomial can be brought to a diagonal form under pre- and postmultiplication by unimodular matrix polynomials, is proved for such polynomials. In contrast to matrix polynomials with real or complex coefficients, a Smith form is generally not unique. For matrix polynomials of first degree, a Kronecker form—the canonical form under strict equivalence—is available, which this chapter presents with a complete proof. Furthermore, the chapter gives a comparison for the Kronecker forms of complex or real matrix polynomials with the Kronecker forms of such matrix polynomials under strict equivalence using quaternion matrices.

This chapter is concerned with the case when both matrices A and B are hermitian. Full and detailed proofs of the canonical forms under strict equivalence and simultaneous congruence are provided, based on the Kronecker form of the pencil A + tB. Several variations of the canonical forms are included as well. Among applications here are: the criteria for existence of a nontrivial positive semidefinite real linear combination and sufficient conditions for simultaneous diagonalizability of two hermitian matrices under simultaneous congruence. A comparison is made with pencils of real symmetric or complex hermitian matrices. It turns out that two pencils of real symmetric matrices are simultaneously congruent over the reals if and only if they are simultaneously congruent over the quaternions. An analogous statement holds true for two pencils of complex hermitian matrices.

This chapter develops diagonal canonical forms and proves inertia theorems for hermitian and skewhermitian matrices with respect to involutions (including the conjugation). These canonical forms enable the identification of maximal neutral and maximal semidefinite subspaces, with respect to a given hermitian or skewhermitian matrix, in terms of their dimensions, which the chapter also discusses. The chapter begins this discussion by introducing canonical forms under congruence, defined as two matrices A,B - ∈ H^n×n, which are said to be congruent if A = S*BS for some invertible S - ∈ H^n×n. If φ‎ is a nonstandard involution, then A,B - ∈ H^n×n are φ‎-congruent if A = SᵩBS for some invertible S.

This chapter extends many of the results from Chapter 29 on single trait-fitness associations to the multiple trait setting. It examines the estimate of multivariate fitness surfaces, starting with quadratic surfaces and then moving to nonparametric versions (which assume no a prior functional form). It also examines path analysis, the analysis of missing data, and multilevel selection.