Unravel Matrix Orthogonality with Ease
Simply input your matrix and let our tool determine if it's orthogonal.
Orthogonal Matrix Checker
Enter your square matrix below to check if it is orthogonal. Separate rows with semicolons (;) and numbers in each row with spaces.
Example: 1 0; 0 1 or cos(θ) -sin(θ); sin(θ) cos(θ)
Input Matrix (A):
Result
Is the matrix orthogonal?
Understanding Orthogonality
A matrix is orthogonal if its transpose is equal to its inverse. Mathematically, this is represented as:
$$A^T = A^{-1}$$
Equivalently, a matrix \(A\) is orthogonal if the product of the matrix and its transpose is the identity matrix \(I\):
$$AA^T = A^TA = I$$
Yes, the given matrix is orthogonal!
This means its columns (and rows) form an orthonormal set of vectors. Orthogonal matrices are crucial in various areas of mathematics and physics, especially in transformations that preserve lengths and angles, such as rotations and reflections.
No, the given matrix is not orthogonal.
This indicates that its columns (or rows) do not form an orthonormal set. The properties of orthogonal matrices, such as preserving lengths and angles, do not apply to this matrix.
What is an Orthogonal Matrix?
In linear algebra, an orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors. Being orthonormal means that the vectors are mutually perpendicular (orthogonal) and normalized (unit length). Orthogonal matrices are essential in various transformations like rotations and reflections because they preserve lengths and angles.
- Orthonormal Vectors: Columns and rows are mutually perpendicular unit vectors.
- Transpose is Inverse: The transpose of an orthogonal matrix is equal to its inverse.
- Preserves Lengths and Angles: Transformations by orthogonal matrices do not change vector lengths or angles between vectors.
- Determinant is ±1: The determinant of an orthogonal matrix is either +1 or -1.
Orthogonal matrices are widely used in computer graphics, signal processing, and physics for rotations, reflections, and coordinate transformations. For further reading, you can explore resources on linear algebra and matrix theory.