Why are orthogonal matrices rotation matrices?

Given a basis of the linear space ℝ3, the association between a linear map and its matrix is one-to-one. A matrix with this property is called orthogonal. So, a rotation gives rise to a unique orthogonal matrix. Thus, an orthogonal matrix leads to a unique rotation.

What is the difference between unitary and orthogonal matrix?

A unitary matrix is a complex matrix such that its conjugate transpose is equal to its inverse. A orthogonal matrix is a squarr matrix such that its transpose is equal to its inverse.

Are reflections orthogonal matrices?

Examples of orthogonal matrices are rotation matrices and reflection matrices. These two types are the only 2 × 2 matrices which are orthogonal: the first column vector has as a unit vector have the form [cos(t),sin(t)]T . The second one, being orthogonal has then two possible directions.

How do orthogonal and unitary matrices relate?

For real matrices, unitary is the same as orthogonal. The rows of a unitary matrix are a unitary basis. That is, each row has length one, and their Hermitian inner product is zero. Similarly, the columns are also a unitary basis.

Are rotation matrices unitary?

If you think about rotations and reflection transformations, they also preserve lengths and distances, so their matrices should indeed be unitary.

What do oblique and orthogonal refer to?

Orthogonal and oblique are two different types of rotation methods used to analyze information from a factor analysis. Rotations where factors are not correlated are orthogonal. So rotation methods that are correlated are oblique while rotation of uncorrelated factors is orthogonal.

What is an orthogonal matrix?

So, basically, orthogonal matrix is just a combination of one-dimensional reflectors and rotations written in appropriately chosen orthonormal basis (the coordinate system you’re used to, but possibly rotated). Fun fact: All orthogonal matrices (even rotations) of order n can be presented as compositions of at most n reflectors.

Is orthogonal matrix isometry of Euclidean space?

Orthogonal matrices also act as an isometry of Euclidean space. Euclidean space is a two dimensional or three dimensional space in which Euclid’s axioms and postulates are valid. A few examples of Euclidean space are reflection, rotation and rotoreflection.

Is the determinant of an orthogonal matrix invertible?

All the orthogonal matrices are invertible. Since the transpose holds back determinant, therefore we can say, determinant of an orthogonal matrix is always equal to the -1 or +1. All orthogonal matrices are square matrices but not all square matrices are orthogonal.

Are rotations and reflections linear orthogonal transformations?

As a particular case, rotations and reflections are linear orthogonal transformations. Since you are interested in visualizations (sorry, I have no reference recommendation), I’ll try to explain it that way. There is something called the eigenvalue decomposition of a matrix, and it is very much related to the Schur decomposition.