QR Decomposition Calculator

Compute a 2×2 QR decomposition using Gram-Schmidt on the matrix columns.

Linear AlgebraMatricesCollege Math

Calculator

a₁₁
a₁₂
a₂₁
a₂₂
Please enter valid values.

What this calculator teaches

QR decomposition factors a matrix into an orthonormal matrix Q and an upper triangular matrix R.

It is widely used in numerical linear algebra and least-squares problems.

Formula & Symbols

ConceptFormulaMeaning
QR factorizationA=QRQ has orthonormal columns, R is upper triangular.
Gram-Schmidtq₂=(v₂-proj_q₁v₂)/||v₂-proj_q₁v₂||Builds orthonormal directions.

Step-by-Step Examples

Example 1
A=[[1,1],[1,0]]
  • Normalize first column (1,1).
  • Remove its projection from second column.
  • Normalize the remainder.
✓ The calculator reports Q and R numerically.

Where students use this

🎓
College mathematics
Use the tool to connect notation, formulas, and numeric answers.
🧪
Engineering analysis
Apply the result to models, systems, approximations, and design calculations.
💻
Computer science
Use matrix, vector, numerical, and optimization ideas in algorithms and data work.

Common Mistakes to Avoid

⚠️
Ignoring restrictions

Check denominator, domain, endpoint, and matrix-invertibility conditions before trusting a result.

⚠️
Rounding too early

Carry extra decimals through intermediate steps and round only the final answer.

⚠️
Mixing symbolic and numeric answers

Some topics need exact symbolic reasoning, while this calculator gives a clear numerical learning check.

Frequently Asked Questions

Is this calculator meant for students? ▾
Yes. It gives an answer and shows the main method so students can learn the reasoning instead of copying only the result.
Can I use decimals and negative numbers? ▾
Yes, when the topic allows them. Some formulas have restrictions such as positive radii, nonzero denominators, or valid intervals.
Why can textbook answers look different? ▾
Equivalent algebraic forms, different rounding, or using exact notation instead of decimals can make answers look different while meaning the same thing.

Related Math Calculators

🔢 Gram-Schmidt Calculator🔢 LU Decomposition Calculator🔢 Least-Squares Solution Calculator

Formula Explorer connections

Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.

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