| Summary: | In this work the matrix exponential function is solved analytically for the special orthogonal groups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>9</mn></mrow></semantics></math></inline-formula>. The number of occurring <i>k</i>-th matrix powers gets limited to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm <i>V</i> and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>, a quadratic equation needs to be solved for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></semantics></math></inline-formula>, a cubic equation for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>6</mn><mo>,</mo><mn>7</mn></mrow></semantics></math></inline-formula>, and a quartic equation for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>8</mn><mo>,</mo><mn>9</mn></mrow></semantics></math></inline-formula>. As an interesting subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>7</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the exceptional Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula> of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The traces of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.
|
|---|