# Topics in Quaternion Linear Algebra by Leiba Rodman

By Leiba Rodman

Quaternions are a host method that has turn into more and more necessary for representing the rotations of gadgets in three-d area and has very important functions in theoretical and utilized arithmetic, physics, computing device technology, and engineering. this is often the 1st publication to supply a scientific, available, and self-contained exposition of quaternion linear algebra. It positive factors formerly unpublished learn effects with entire proofs and plenty of open difficulties at quite a few degrees, in addition to greater than 2 hundred routines to facilitate use by way of scholars and teachers. purposes awarded within the publication comprise numerical levels, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.

Designed for researchers and scholars throughout numerous disciplines, the publication should be learn by way of an individual with a history in linear algebra, rudimentary advanced research, and a few multivariable calculus. teachers will locate it priceless as a complementary textual content for undergraduate linear algebra classes or as a foundation for a graduate path in linear algebra. The open difficulties can function study tasks for undergraduates, issues for graduate scholars, or difficulties to be tackled by means of expert learn mathematicians. The publication is additionally a useful reference software for researchers in fields the place strategies in response to quaternion research are used.

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Up are linearly independent. Let a1 χn,p ([u1 · · · up ]) ... = 0 ap for some a1 , . . , ap ∈ R4×1 , and, in turn, aj0 aj1 aj = −aj3 , aj2 aj ∈ R, = 0, 1, 2, 3. 1) 35 VECTOR SPACES AND MATRICES: BASIC THEORY Write uj1 uj = ... 2) and, in turn, (0) (1) (2) (3) ujk = ujk + ujk i + ujk j + ujk k, ( ) ujk ∈ R, = 0, 1, 2, 3. ,p Xj = [X1 X2 . . Xp ]. ,p (0) ujk (1) ujk −u(3) jk (2) ujk (1) −ujk (0) ujk (2) ujk (3) ujk (3) ujk (2) −ujk (0) ujk (1) ujk (2) −ujk (3) −ujk (1) −ujk (0) ujk a10 a11 −a13 a12 a20 a21 −a23 a22 ..

1. 2. Ex. 2. 4. Ex. 3. Prove that |x + y| = |x| + |y|, x, y ∈ H, holds if and only if either at least one of x, y is zero or x = 0 and y = 0 are positive real multiples of each other. Ex. 4. Solve the following equations: (1) x4 + 1 = 0, x ∈ H; (2) xm + 1 = 0, x ∈ H, where m is a fixed even positive integer. Hint: For a fixed q ∈ H, with R(q) = 0 and |q| = 1, consider solutions in SpanR {1, q} which is isomorphic to C. Ex. 5. Verify the following equality for x ∈ H: x2 = |R(x)|2 − |V(x)|2 + 2R(x)V(x).

Let A ∈ Hm×n . , uniquely determined by A; (b) A = A0 Y , where Y ∈ Hn×n is invertible and A0 ∈ Hm×n is a column reduced echelon form; moreover, A0 is unique; (c) (QR factorization) if m = n, then A = QR, where Q is unitary and R is upper triangular with nonnegative diagonal elements; moreover, if A is invertible, then Q and R are unique; (d) (polar decomposition) if m = n, then A = RU , where R is positive semidefinite and U is unitary; moreover, if A is invertible, then R and U are unique; (e) (rank decompositions) if rank (A) = k = 0, then A = BC, where B ∈ Hm×k , C ∈ Hk×n ; also, Ik 0 A=B C, 0 0(m−k)×(n−k) where B ∈ Hm×m , C ∈ Hn×n are invertible; (f) (singular value decomposition) if A = 0, then there exist unitary U ∈ Hm×m , V ∈ Hn×n , and real positive numbers a1 ≥ a2 ≥ · · · ≥ ak , where k = rank (A), such that diag (a1 , .