This tool calculates the orthogonal complement of a given vector space for you.

## Orthogonal Complement Calculator

This calculator computes the orthogonal complement of a given matrix. An orthogonal complement of a subspace is the set of all vectors that are orthogonal to each vector in the subspace.

### How to Use

- Select the matrix size you want to use (2×2, 3×3, or 4×4).
- Input the values of the matrix in the provided fields.
- Click the “Calculate” button to get the orthogonal complement of the matrix.

### How It Calculates

The calculator uses the Gram-Schmidt process to find the orthogonal complement. It iteratively adjusts a basis to be orthogonal to the matrix you have provided. It performs normalization of the resulting vectors to ensure the complement is accurately calculated.

### Limitations

This calculator can handle matrices up to 4×4. For matrices larger than this, you will need more advanced software or tools for calculation due to complexity. Additionally, the precision of floating-point arithmetic may result in minor inaccuracies for highly degenerate matrices.

## Use Cases for This Calculator

### Calculating Orthogonal Complement Use Cases

Calculate the orthogonal complement of a vector: Enter the vector in the input field, and with the click of a button, you’ll get the orthogonal complement of the vector displayed.

Compute orthogonal complement of a vector space: Input the basis of the vector space, and the tool will find the orthogonal complement space for you, helping you in your linear algebra computations.

Verify if two vectors are orthogonal: Enter the two vectors in the designated fields, and the calculator will determine if the vectors are orthogonal to each other, simplifying vector operations for you.

Determine if a set of vectors is orthogonal: Input the vectors in the provided area, and the tool will analyze if they form an orthogonal set, aiding you in your vector space analysis.

Calculate the orthogonal projection of a vector: Enter the vector and the basis vector to find the orthogonal projection, assisting you in various mathematical calculations.

Find the orthogonal complement of a subspace: Input the subspace basis, and the calculator will compute the orthogonal complement, aiding in your linear algebra problem-solving process.

Check if a matrix is orthogonal: Enter the matrix, and with a quick click, the tool will determine if the matrix is orthogonal, helping you in matrix operations.

Compute the orthogonal decomposition of a vector: Provide the vector and the basis, and the calculator will illustrate the orthogonal decomposition of the vector, making your vector transformations easier.

Determine the orthogonal left inverse of a matrix: Input the matrix, and the tool will find the orthogonal left inverse if it exists, aiding you in your matrix calculations.

Verify if two subspaces are orthogonal: Input the basis vectors of the subspaces, and the calculator will check if the subspaces are orthogonal to each other, simplifying subspace analyses for you.