Foundational aspects of a new matrix holomorphic structure

Purpose In this paper we talk about complex matrix quaternions (biquaternions) and we deal with some abstract methods in mathematical complex matrix analysis. Design/methodology/approach We introduce and investigate the complex space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi mathvariant="double-struck">H</m:mi></m:mrow><m:mrow><m:mi mathvariant="double-struck">C</m:mi></m:mrow></m:msub></m:math> consisting of all 2 × 2 complex matrices of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mspace width="0.28em" /><m:mi>ξ</m:mi><m:mo>=</m:mo><m:mfenced open="(" close=")"><m:mrow><m:mtable class="matrix"><m:mtr><m:mtd columnalign="center"><m:msub><m:mrow><m:mi>z</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo>+</m:mo><m:mi>i</m:mi><m:msub><m:mrow><m:mi>w</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub></m:mtd><m:mtd columnalign="center"><m:msub><m:mrow><m:mi>z</m:mi></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:msub><m:mo>+</m:mo><m:mi>i</m:mi><m:msub><m:mrow><m:mi>w</m:mi></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:msub></m:mtd></m:mtr><m:mtr><m:mtd columnalign="center"><m:mo>−</m:mo><m:msub><m:mrow><m:mover accent="true"><m:mrow><m:mi>z</m:mi></m:mrow><m:mo>‾</m:mo></m:mover></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:msub><m:mo>−</m:mo><m:mi>i</m:mi><m:msub><m:mrow><m:mover accent="true"><m:mrow><m:mi>w</m:mi></m:mrow><m:mo>‾</m:mo></m:mover></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:msub></m:mtd><m:mtd columnalign="center"><m:msub><m:mrow><m:mover accent="true"><m:mrow><m:mi>z</m:mi></m:mrow><m:mo>‾</m:mo></m:mover></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo>+</m:mo><m:mi>i</m:mi><m:msub><m:mrow><m:mover accent="true"><m:mrow><m:mi>w</m:mi></m:mrow><m:mo>‾</m:mo></m:mover></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub></m:mtd></m:mtr></m:mtable></m:mrow></m:mfenced></m:math>, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:mrow><m:msub><m:mrow><m:mi>z</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:msub><m:mrow><m:mi>w</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:msub><m:mrow><m:mi>z</m:mi></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:msub><m:mo>,</m:mo><m:msub><m:mrow><m:mi>w</m:mi></m:mrow><m:mrow><m:mn>2</m:mn></m:mrow></m:msub></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>∈</m:mo><m:msup><m:mrow><m:mi mathvariant="double-struck">C</m:mi></m:mrow><m:mrow><m:mn>4</m:mn></m:mrow></m:msup></m:math> . Findings We develop on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi mathvariant="double-struck">H</m:mi></m:mrow><m:mrow><m:mi mathvariant="double-struck">C</m:mi></m:mrow></m:msub></m:math> a new matrix holomorphic structure for which we provide the fundamental operational calculus properties. Originality/value We give sufficient and necessary conditions in terms of Cauchy–Riemann type quaternionic differential equations for holomorphicity of a function of one complex matrix variable <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ξ</m:mi><m:mo>∈</m:mo><m:msub><m:mrow><m:mi mathvariant="double-struck">H</m:mi></m:mrow><m:mrow><m:mi mathvariant="double-struck">C</m:mi></m:mrow></m:msub></m:math> . In particular, we show that we have a lot of holomorphic functions of one matrix quaternion variable.