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Orthogonal and unitary matrices are very special types of matrices. Imagine them as "clean" transformations that don't change distances and angles in space, only rotating or reflecting objects.
The difference is simple. Orthogonal matrices work with real numbers, while unitary matrices work with complex numbers. Both have the same properties, just different versions.
A square real matrix is called orthogonal if:
This means, to get the inverse of this matrix, we just transpose it. Very practical, right?
This is equivalent to:
A square complex matrix is called unitary if:
Here is the conjugate transpose of . The concept is similar, just for complex numbers.
This is also equivalent to:
Real orthogonal matrices are actually a special case of unitary matrices, since .
What's interesting about orthogonal and unitary matrices is that their determinants always have absolute value . Why is this?
For a unitary matrix , we have . If we calculate its determinant:
So . For orthogonal matrices, the proof is the same, just using .
Eigenvalues of orthogonal and unitary matrices also have special properties. Every eigenvalue always satisfies:
Why is this? Suppose for an eigenvector . For the complex case, we can calculate:
Since , then , so .
For real orthogonal matrices, the eigenvalues can be or if real. But if complex, they can be written as:
This means complex eigenvalues lie on the unit circle in the complex plane.
Let's look at a familiar example, the rotation matrix:
We can check that this is an orthogonal matrix:
The characteristic polynomial is:
The eigenvalues are:
The result is .
The transformation represents a rotation by angle . For and , this matrix has no real eigenvalues, but has two complex eigenvalues.
