Q:
# What is an orthonormal matrix?

**An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors or orthonormal vectors.** Similarly, a matrix Q is orthogonal if its transpose is equal to its inverse.

Credit:
Andrey Prokhorov
E+
Getty Images

An orthogonal matrix Q is necessarily square and invertible with inverse Q^{−1} = Q^{T}. As a linear transformation, an orthogonal matrix preserves the dot product of vectors and therefore acts as an isometry of Euclidean space. In other words, it is a unitary transformation.

The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n), consisting of orthogonal matrices with determinant +1, is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.

The most common examples of orthogonal matrices are rotations and reflections. Both are important for developing numerical methods. The fact that orthogonal matrices don't change the lengths of vectors makes them very desirable in numerical applications since they do not increase rounding errors significantly. They are therefore favored for stable algorithms.They are particularly important for eigenvalue methods for symmetric matrices because they produce similarity transforms that preserve the symmetric property.

Learn more about Arithmetic-
Q:
## How do you calculate an inverse matrix in Matlab?

A: It is easy to find the inverse of a matrix in MATLAB. Input the matrix, then use MATLAB's built-in inv() command to get the inverse.... Full Answer >Filed Under: -
Q:
## What is matrix division?

A: Matrix division means multiplying one matrix by the inverse of another matrix. Matrix A "divided by" matrix B is equivalent to matrix A multiplied by the i... Full Answer >Filed Under: -
Q:
## What are some examples of 5th grade math problems?

A: Fifth grade children learn to add and subtract large numbers by writing them in columns rather than horizontally. These multiplication problems can be writ... Full Answer >Filed Under: -
Q:
## Can you plant 10 trees in five rows with four trees in each row?

A: Planting 10 trees in five rows yields two trees per row. Putting four trees in a row yields two full rows with two remaining trees. An optimal pattern for ... Full Answer >Filed Under: