## 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:

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.