Orthogonal Matrix Checker

Math Linear Algebra • Applications and Advanced Linear Algebra (capstone)

Written by STEM Calculators Team

Check whether a square matrix \(Q\) is orthogonal by testing \(Q^{\mathsf T}Q=I\). Also compute an orthonormal basis from the columns of \(Q\) using the full Gram–Schmidt process and return a QR-style output.

Matrix size

Tolerance

Display decimals

Orthonormalization method

Matrix \(Q\)

Columns are treated as the vectors to orthonormalize.

Inputs accept -3.5, 2e-4, fractions 7/3, and constants pi, e.

Ready

Results

Orthogonal?

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\(\lVert Q^{\mathsf T}Q-I\rVert_F\)

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\(\det(Q)\)

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Interpretation

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\(Q^{\mathsf T}Q\)

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\(\Delta = Q^{\mathsf T}Q - I\)

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Gram–Schmidt orthonormal matrix \(\hat{Q}\)

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Upper triangular \(R\) (QR link: \(Q \approx \hat{Q}R\))

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If \(Q\) is orthogonal, then \(Q^{-1}=Q^{\mathsf T}\). If \(\det(Q)\approx 1\), it behaves like a rotation; if \(\det(Q)\approx -1\), a reflection.

Graph

2×2 only. Zoom with wheel/trackpad, drag to pan, double-click to reset view. Click Play to animate Gram–Schmidt.

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Step-by-step

Enter \(Q\) and click “Calculate”.

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Theory

Orthogonal matrices

A square matrix \(Q\in\mathbb{R}^{n\times n}\) is orthogonal if its columns form an orthonormal set. Algebraically, this is equivalent to the identity \[ Q^{\mathsf T}Q = I. \] When \(Q\) is orthogonal, it preserves lengths and angles: \(\lVert Qx\rVert = \lVert x\rVert\) and \((Qx)\cdot(Qy)=x\cdot y\). This is why orthogonal matrices represent rigid transformations such as rotations and reflections (in higher dimensions, “rigid motion” generalizations).

Two immediate consequences are: \[ Q^{-1} = Q^{\mathsf T} \quad\text{and}\quad |\det(Q)| = 1. \] If \(\det(Q)=1\), the transformation is rotation-like (orientation-preserving). If \(\det(Q)=-1\), it is reflection-like (orientation-reversing). In numerical computations, we rarely get exact equalities; instead we measure how close \(Q^{\mathsf T}Q\) is to \(I\), for example with the Frobenius norm \(\lVert Q^{\mathsf T}Q - I\rVert_F\).

Gram–Schmidt orthonormalization

Gram–Schmidt converts a set of linearly independent vectors \(\{a_1,\dots,a_n\}\) into an orthonormal set \(\{q_1,\dots,q_n\}\) that spans the same subspace. In matrix form, you can think of the columns of \(A\) as the input vectors: \(A=[a_1\ a_2\ \cdots\ a_n]\). The classical Gram–Schmidt steps are:

\[ \begin{aligned} v_1 &= a_1, \qquad q_1 = \frac{v_1}{\lVert v_1\rVert},\\ v_j &= a_j - \sum_{i=1}^{j-1}(q_i^{\mathsf T}a_j)\,q_i,\qquad q_j = \frac{v_j}{\lVert v_j\rVert}\quad (j\ge 2). \end{aligned} \]

The term \((q_i^{\mathsf T}a_j)q_i\) is the projection of \(a_j\) onto the direction \(q_i\). Subtracting all these projections removes the components of \(a_j\) that lie in the span of the previous orthonormal vectors, leaving a vector \(v_j\) orthogonal to the earlier \(q_i\)’s. Normalizing \(v_j\) produces a unit vector.

If some \(v_j\) becomes (nearly) the zero vector, that means \(a_j\) was dependent on earlier columns, so it does not add a new direction to the span. Practical implementations detect this using a tolerance and report that the set is not fully independent.

QR connection and numerical stability

Gram–Schmidt naturally produces a QR factorization. Let \(\hat{Q}\) be the matrix whose columns are the orthonormal vectors \(q_1,\dots,q_n\). Define \(R\) by \(r_{ij}=q_i^{\mathsf T}a_j\) for \(i\le j\) and \(r_{ij}=0\) for \(i>j\). Then, when the columns are independent, \[ A = \hat{Q}R, \] where \(\hat{Q}\) is orthonormal and \(R\) is upper triangular. This is a cornerstone of least squares, eigenvalue algorithms, and many numerical linear algebra methods.

In floating-point arithmetic, classical Gram–Schmidt can lose orthogonality when vectors are nearly dependent. Modified Gram–Schmidt improves stability by re-projecting in a different order. At the university level, Householder reflections are often used for the most stable QR factorization, especially for large matrices or ill-conditioned problems.

Frequently Asked Questions

What does it mean for a matrix to be orthogonal?

A matrix Q is orthogonal if Q^T Q = I, meaning its columns are orthonormal and it preserves lengths and angles.

Why can Q^T Q be close to I but not exactly I?

Floating-point rounding and entered decimals make exact equality unlikely. The calculator uses a tolerance and reports ||Q^T Q − I||_F.

What does det(Q) tell me?

If Q is orthogonal, det(Q) is approximately ±1. det(Q)≈1 suggests rotation-like behavior, while det(Q)≈−1 suggests reflection-like behavior.

Why can Gram–Schmidt produce a zero vector?

That happens when a column is linearly dependent on previous columns. The algorithm detects near-zero norms and marks the set as dependent.

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Frequently Asked Questions

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