Understand the subject
What is a projection matrix?
A projection keeps the component inside a chosen subspace and discards the perpendicular remainder.
Use it to project a vector or model output onto an independent column space.
See the structure
What the calculation is doing
input vector× Aoutput vectorP=A(AᵀA)⁻¹Aᵀ
Worked interpretation
Read the result in context
P=A(AᵀA)⁻¹Aᵀ
Interpret with care
Important boundary
The entered spanning columns must be independent for this focused formula.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
- Set the matrix dimensions required by the task.
- Fill every visible entry and any operation-specific control.
- Read the result type, factors, classification, and disclosed limitations.
Calculation method
Apply the stated linear-algebra contract
P = A(AᵀA)⁻¹Aᵀ
Undefined, singular, dimensionally incompatible, dependent, or unsupported inputs receive an explicit message instead of a fabricated numeric result.
Worked example
Example interpretation
The result satisfies P²=P and Pᵀ=P.
P = A(AᵀA)⁻¹Aᵀ
Supported inputs
Precision and limits
Matrix size
Input matrices are limited to 1–6 rows and columns; operation-specific square, rank, or compatibility rules still apply.
Entry precision
Enter signed finite decimals with at most 30 digits and 15 decimal places.
Exactness
Finite decimals become reduced rational numbers and intermediate structural calculations are not rounded.
Supported scope
The entered spanning columns must be linearly independent.
