Introduction to Matrix Groups

Examples of Matrix Groups

The Orthogonal Group

The orthogonal group is the group of square matrices that satisfies one particular property. The square matrix A is an orthogonal matrix if A multiplied by its own transpose produces the identity matrix, i.e. $$AA^T=I$$. Because a matrix multiplied by its inverse always equals the identity matrix, every orthogonal matrix, A, has the same transpose and inverse, i.e. $$A^{-1}=A^T$$.

The Unitary Group

The unitary matrices are very similar to the orthogonal matrices. There is one main difference between the two. The defining characteristic of a unitary matrix is that the inverse is equal not to the transpose, but to the conjugate transpose. This means that for a matrix B, B multiplied by its conjugate transpose will equal the identity.

Click on green terms to see definitions.

The transpose of a matrix is the same matrix but with the values flipped around the diagonal from the top-left corner to the bottom right corner. For example:

\begin{eqnarray} A = \left[ \matrix{a & b & c \\ d & e & f \\ g & h & i} \right] \\ A^T = \left[ \matrix{ a & d & g \\ b & e & h \\ c & f & i} \right] \end{eqnarray}

The conjugate of a matrix is the matrix wich each value replaced by the conjugate of that value. Every number cna be represented as a complex number with real and imaginary parts as $$a+bi$$. For any given $$a+bi$$, its conjugate is $$a-bi$$. If a number has no imaginary part, then $$b=0$$ and the conjugate is the same number. The transpose conjugate is simply the transpose of the conjugate matrix. It is also possible ot create the transpose conjugate by first taking the transpose and then taking the conjugate.