QR Decomposition Calculator

How QR Decomposition Calculator Works?

The QR Decomposition Calculator accepts the nine elements of a real 3×3 matrix $A$ and separates it into an orthogonal matrix $Q$ and an upper triangular matrix $R$ such that $A=QR$. The input matrix is represented as $A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$.

The calculation orthogonalizes the columns of $A$. If the columns are $a_1$, $a_2$, and $a_3$, the first normalized vector is formed by dividing the first column by its Euclidean length.

Each later column is made orthogonal to the previously calculated vectors by subtracting projections. The coefficients of these projections become the off-diagonal entries of $R$.

The resulting matrices have the forms shown below. The nine Q outputs are the entries of the orthogonal matrix, while the nine R outputs include zeros below the main diagonal.

The determinant is obtained from the factors using $\det \left(A\right)=\det \left(Q\right)\det \left(R\right)$. Because $R$ is triangular, its determinant is the product of its diagonal entries, so $\det \left(A\right)=\det \left(Q\right)r_{11}{r}_{22}{r}_{33}$. The orthogonality error checks how close $Q^{T}Q$ is to the identity matrix, while the reconstruction error checks how close the product $QR$ is to the original matrix $A$.

Practical Use Cases

Students and instructors can enter a 3×3 textbook matrix and inspect every element of Q and R, making Gram–Schmidt orthogonalization, triangular structure, determinants, and reconstruction checks easier to verify.

Engineers can use QR factors to analyze systems of linear equations, examine matrix conditioning, and check whether a computed decomposition reproduces the original design or measurement matrix.

Researchers and data scientists can use the factors as a compact numerical foundation for least-squares calculations, where the orthogonal transformation represented by Q helps preserve vector lengths and reduce computational complications.

Numerical-methods developers can compare the orthogonality error and reconstruction error against expected floating-point tolerances when testing matrix algorithms or validating sample implementations.

Applied mathematicians can use the determinant output together with the triangular factor to assess singularity, rank-related behavior, and the orientation encoded by the orthogonal factor.

Accuracy and Limitations

The calculator uses finite-precision numerical arithmetic, so Q may satisfy $Q^{T}Q=I$ and $A=QR$ only approximately; the reported orthogonality error and reconstruction error quantify those effects. Results can become less reliable when columns are linearly dependent or nearly dependent, because a diagonal entry of R approaches zero and normalization becomes sensitive to rounding. Inputs are limited to the provided range from −100000 to 100000 with 0.01 increments, and the calculator handles only real 3×3 matrices, not rectangular, complex, or larger matrices. Determinants and individual factor entries may also lose relative accuracy for poorly conditioned matrices even when the displayed residual is small.

Quick Examples