Calculation Result
This result appears above the form after you calculate.
Input Matrix Q
QᵀQ
QᵀQ - I
Plotly Graph
The heatmap shows how far QᵀQ deviates from the identity matrix.
Calculator Form
Use the responsive grid below. Large screens show three columns, smaller screens show two, and mobile shows one.
Example Data Table
| Example | Matrix | Orthogonal? | Determinant | Interpretation |
|---|---|---|---|---|
| Identity | [1 0] [0 1] | Yes | 1 | Special orthogonal and orientation preserving. |
| Rotation by 90° | [0 -1] [1 0] | Yes | 1 | Pure rotation. Lengths and angles stay unchanged. |
| Reflection in x-axis | [1 0] [0 -1] | Yes | -1 | Orthogonal but orientation reversing. |
| Permutation matrix | [0 1 0] [0 0 1] [1 0 0] | Yes | 1 | Reorders basis vectors without stretching them. |
Formula Used
For a real square matrix Q, orthogonality means:
QᵀQ = I
Its transpose equals its inverse:
Q⁻¹ = Qᵀ
Take determinants on both sides of
QᵀQ = I:
det(QᵀQ) = det(I) = 1
Since
det(Qᵀ) = det(Q), then:
det(Q)² = 1
Therefore:
det(Q) = 1 or det(Q) = -1
The calculator forms the deviation matrix:
E = QᵀQ - I
It then reports:
max |Eᵢⱼ|
If this maximum error is within the chosen tolerance, the matrix is treated as orthogonal.
The determinant is computed numerically using row reduction with partial pivoting. This is stable for general square matrices and helps handle larger inputs more reliably than direct cofactor expansion.
How to Use This Calculator
- Select the matrix size you want to analyze.
- Either enter values manually, apply a preset, or paste matrix rows.
- Choose a tolerance for the orthogonality check.
- Set the number of displayed decimals for cleaner output.
- Click the calculation button to show the result above the form.
- Review the determinant, orthogonality status, error matrix, and graph.
- Use the download buttons to export a CSV or PDF report.
Frequently Asked Questions
1) What is an orthogonal matrix?
An orthogonal matrix has rows and columns that are mutually perpendicular unit vectors. It satisfies QᵀQ = I, so its inverse equals its transpose. Such matrices preserve lengths, angles, and inner products.
2) How to determine if matrix is orthogonal?
Compute QᵀQ and compare it with the identity matrix. If diagonal entries are near 1 and off-diagonal entries are near 0 within a chosen tolerance, the matrix is orthogonal. This calculator reports the maximum deviation.
3) Prove that the determinant of an orthogonal matrix is equal to 1 or −1.
Start from QᵀQ = I. Taking determinants gives det(QᵀQ) = det(I) = 1. Because det(Qᵀ) = det(Q), we get det(Q)² = 1. Therefore det(Q) must be 1 or −1.
4) Prove that the determinant of an orthogonal matrix is 1.
That is not true for every orthogonal matrix. Reflections have determinant −1. You can only prove determinant 1 for special orthogonal matrices, such as pure rotations, which preserve orientation as well as lengths and angles.
5) What does determinant −1 mean for an orthogonal matrix?
A determinant of −1 means the transformation reverses orientation. Geometrically, this often indicates a reflection or a reflection combined with a rotation. The matrix still preserves lengths and angles, so it remains orthogonal.
6) Can a non-orthogonal matrix still have determinant 1 or −1?
Yes. Determinant alone does not prove orthogonality. A matrix may have determinant 1 or −1 and still fail the condition QᵀQ = I. That is why this calculator checks both determinant and orthogonality error.
7) Why does the calculator use tolerance instead of exact equality?
Decimal inputs and floating-point arithmetic introduce tiny rounding errors. Tolerance lets the calculator decide whether a matrix is effectively orthogonal in practice, even when values are extremely close rather than exactly perfect.
8) What matrix sizes does this page support?
This version supports square matrices from 2 × 2 up to 6 × 6. That range works well for manual entry, classroom use, and common examples while keeping the interface readable and the calculations fast.