This tool will help you determine if a matrix is orthogonal.

**How to use the orthogonal matrix calculator:**

To use this calculator, input a 3×3 matrix into the provided text area. Separate the elements of each row by a comma and each row by a newline.

Click the “Calculate” button to determine if the matrix is orthogonal. The result will be displayed in the result input field.

**How it works:**

The calculator checks if the matrix is orthogonal by first transposing the matrix. Next, it multiplies the original matrix with the transposed one. Finally, it checks if the resultant matrix is an identity matrix. If the resultant matrix is an identity matrix, the original matrix is orthogonal.

**Limitations:**

This calculator only handles 3×3 matrices. The input must be numeric and correctly formatted for the calculator to work. Invalid inputs will generate an error message indicating that the input is invalid.

## Use Cases for This Calculator

### Calculate Determinant of an Orthogonal Matrix

Enter the values of a 3×3 orthogonal matrix and instantly find the determinant. Obtain the result in a single click to determine the overall scaling factor of the matrix, which indicates how it affects the area or volume it operates on.

### Check if Matrix is Orthogonal

Input any square matrix to verify if it is orthogonal using the calculator. Confirm if the matrix columns are orthonormal vectors to ensure that the matrix has a valid rotation or reflection operation.

### Find Inverse of an Orthogonal Matrix

Submit the orthogonal matrix values and obtain the inverse matrix immediately. Use this feature to efficiently compute reflections or rotations, reversing the transformation operation of the original matrix.

### Calculate Transpose of an Orthogonal Matrix

Explore the tool to quickly transpose an orthogonal matrix. Swap the rows and columns to see the effect on the matrix; this enables you to analyze reflections, rotations, or other transformations more effectively.

### Compute Eigenvalues of an Orthogonal Matrix

Enter the matrix elements to determine the eigenvalues directly with ease. Analyze the stability of the matrix transformations by understanding how the values scale the eigenvectors during the transformation process.

### Calculate Trace of an Orthogonal Matrix

Input a square matrix and instantly get the trace value. Utilize this calculation to verify properties of the matrix or understand how the elements affect the main diagonal of the matrix.

### Verify if Matrix is Idempotent

Check any square matrix to see if it is idempotent. Use the calculator to confirm if multiplying the matrix by itself results in the original matrix, assisting in analyzing projection or reflection operations.

### Find Rank of an Orthogonal Matrix

Input a matrix to compute the rank value promptly. Employ this function to determine the dimension of the output space affected by the matrix transformation, serving insights into the impact of the transformation.

### Compute Null Space of an Orthogonal Matrix

Insert the matrix values to calculate the null space efficiently. Understand which vectors are unaffected by the transformation operation of the matrix, providing insights into properties like rotation or reflection symmetry.

### Calculate Spectral Decomposition of an Orthogonal Matrix

Submit the matrix elements to instantly obtain the spectral decomposition. Break down the matrix into eigenvalues and eigenvectors to simplify complex transformations and understand the impact of the matrix on different dimensions.