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coequate

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Definitions of coequate in various dictionaries:

verb - to equate with something else

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Coequate might refer to
In abstract algebra, the Split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, 1/2(1 + j) is an idempotent zero-divisor, and i − j is nilpotent.) As a mathematical structure, they form an algebra over the real numbers, which is isomorphic to the algebra of 2 × 2 real matrices. For other names for split-quaternions see the Synonyms section below. * The set {1, i, j, k} forms a basis. The products of these elements are* ij = k = −ji, * jk = −i = −kj, * ki = j = −ik, * i2 = −1, * j2 = +1, * k2 = +1,and hence ijk = 1. It follows from the defining relations that the set {1, i, j, k, −1, −i, −j, −k} is a group under coquaternion multiplication; it is isomorphic to the dihedral group D4, the symmetry group of a square. * A coquaternion * * q = w + xi + yj + zk, has a conjugate q∗ = w − xi − yj − zk.Due to the anti-commutative property of its basis vectors, the product of a coquaternion with its conjugate is given by an isotropic quadratic form: * * * * * N * ( * q * ) * = * q * * q * * ∗ * * * = * * w * * 2 * * * + * * x * * 2 * * * − * * y * * 2 * * * − * * z * * 2 * * * . * * * {\displaystyle N(q)=qq^{}=w^{2}+x^{2}-y^{2}-z^{2}.} Given two coquaterions p and q, one has N(p q) = N(p) N(q), showing that N is a quadratic form admitting composition. This algebra is a composition algebra and N is its norm. Any q ≠ 0 such that N(q) = 0 is a null vector, and its presence means that coquaternions form a "split composition algebra", and hence a coquaternion is also called a split quaternion. * When the norm is non-zero, then q has a multiplicative inverse, namely q∗/N(q). The set * * U = {q : qq∗ ≠ 0}is the set of units. The set P of all coquaternions forms a ring (P, +, •) with group of units (U, •). The coquaternions with N(q) = 1 form a non-compact topological group SU(1, 1), shown below to be isomorphic to SL(2, R). * Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra. * *

Coequate might refer to

In abstract algebra, the Split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, 1/2(1 + j) is an idempotent zero-divisor, and i − j is nilpotent.) As a mathematical structure, they form an algebra over the real numbers, which is isomorphic to the algebra of 2 × 2 real matrices. For other names for split-quaternions see the Synonyms section below.
* The set {1, i, j, k} forms a basis. The products of these elements are* ij = k = −ji,
* jk = −i = −kj,
* ki = j = −ik,
* i2 = −1,
* j2 = +1,
* k2 = +1,and hence ijk = 1. It follows from the defining relations that the set {1, i, j, k, −1, −i, −j, −k} is a group under coquaternion multiplication; it is isomorphic to the dihedral group D4, the symmetry group of a square.
* A coquaternion
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* q = w + xi + yj + zk, has a conjugate q∗ = w − xi − yj − zk.Due to the anti-commutative property of its basis vectors, the product of a coquaternion with its conjugate is given by an isotropic quadratic form:
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* {\displaystyle N(q)=qq^{*}=w^{2}+x^{2}-y^{2}-z^{2}.}
* Given two coquaterions p and q, one has N(p q) = N(p) N(q), showing that N is a quadratic form admitting composition. This algebra is a composition algebra and N is its norm. Any q ≠ 0 such that N(q) = 0 is a null vector, and its presence means that coquaternions form a "split composition algebra", and hence a coquaternion is also called a split quaternion.
* When the norm is non-zero, then q has a multiplicative inverse, namely q∗/N(q). The set
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* U = {q : qq∗ ≠ 0}is the set of units. The set P of all coquaternions forms a ring (P, +, •) with group of units (U, •). The coquaternions with N(q) = 1 form a non-compact topological group SU(1, 1), shown below to be isomorphic to SL(2, R).
* Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra.
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