Orthogonal Matrix Checker Calculator

Matrix Input

Enter a square matrix

Use spaces or commas between values. Start a new row on each line.

Checker Settings

Numerical tolerance

Displayed decimal places

Orthogonal rule: AᵀA = I and AAᵀ = I

For exact orthogonal matrices, the determinant should be +1 or −1.

Actions

Supported format

Example 3×3 input:
1 0 0
0 0 -1
0 1 0

Example Data Table

Matrix Type	Example Matrix	Expected Result	Reason
2D rotation	[0 -1; 1 0]	Orthogonal	AᵀA equals the identity matrix.
Identity	[1 0; 0 1]	Orthogonal	Rows and columns are unit, perpendicular vectors.
Scaled matrix	[2 0; 0 1]	Not orthogonal	One column norm is not equal to one.
Repeated direction	[1 0; 1 0]	Not orthogonal	Columns are not perpendicular.

Formula Used

An orthogonal matrix satisfies AᵀA = I and equivalently AAᵀ = I. The transpose is also the inverse, so A⁻¹ = Aᵀ.

The calculator evaluates both products, compares them with the identity matrix, and measures the maximum absolute deviation and Frobenius norm error.

It also computes the determinant. For an orthogonal matrix, the determinant magnitude equals one, so det(A) = ±1.

How to Use This Calculator

Enter a square matrix in the input box.
Place each row on a new line.
Separate values using spaces or commas.
Choose a tolerance for floating-point comparisons.
Select the number of displayed decimal places.
Press Check Matrix to compute the result.
Review AᵀA, AAᵀ, determinant, and norm diagnostics.
Use the export buttons to save CSV or PDF outputs.

FAQs

1. What does orthogonal mean for a matrix?

A matrix is orthogonal when its transpose multiplied by itself equals the identity matrix. Its rows and columns form perpendicular unit vectors.

2. Why must the matrix be square?

Only square matrices can equal an identity matrix of matching size. That requirement is necessary for the inverse and transpose relationship.

3. Why does tolerance matter?

Computed matrices often contain rounding noise. Tolerance lets you treat tiny numerical deviations as acceptable when checking near-orthogonal matrices.

4. Does determinant alone prove orthogonality?

No. A determinant of plus or minus one is necessary for orthogonal matrices, but it does not guarantee perpendicular unit rows and columns.

5. What if only AᵀA is close to identity?

For square matrices, both conditions are equivalent theoretically. This calculator checks both products to provide stronger numerical diagnostics and clearer reporting.

6. Can this check rotation matrices?

Yes. Rotation and reflection matrices are classic orthogonal matrices, provided their rows and columns remain unit length and mutually perpendicular.

7. Can I export the calculated results?

Yes. After calculation, use the CSV button for spreadsheet-friendly output or the PDF button for a clean printable report.