Last updated: August 5, 2026
Pseudoinverse Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The Pseudoinverse Calculator computes the Moore-Penrose pseudoinverse of a full-rank 2x1 column vector using A⁺ = Aᵀ / (a1² + a2²). It returns the resulting row vector, the squared norm used in the division, and an identity check confirming A⁺A equals 1, connecting the abstract formula to least-squares applications.
The pseudoinverse of a vector divides its transpose by the sum of its squared entries, giving the best generalized inverse when no true inverse exists.
Key Takeaways
- The Moore-Penrose pseudoinverse generalizes the matrix inverse to rectangular or singular matrices.
- For a 2x1 column vector, AᵀA collapses to a single scalar, making the pseudoinverse easy to compute directly.
- The pseudoinverse of an m×n matrix is always n×m in shape — a column vector's pseudoinverse is a row vector.
- A⁺A equals 1 for any valid, non-zero column vector, confirming the identity-like property.
- The pseudoinverse is the mathematical foundation of least-squares regression and best-fit solutions.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
A = [a1; a2] is 2×1; AᵀA = a1² + a2² is a scalar; A⁺ = Aᵀ / (a1² + a2²), a 1×2 row vector
Where:
- a1=First entry of the column vector
- a2=Second entry of the column vector
- A⁺=Moore-Penrose pseudoinverse of A
Watch & Learn
Gilbert Strang introduces the Moore-Penrose pseudoinverse, explaining how it generalizes matrix inversion to non-square and singular matrices.
Worked Examples
Pseudoinverse of a 3-4-5 style vector
A classic vector whose squared norm is a perfect square, making the arithmetic clean.
- 1Compute the squared norm: 3² + 4² = 9 + 16 = 25.
- 2Divide each entry by the squared norm: 3/25 = 0.12, 4/25 = 0.16.
- 3Result: A⁺ = [0.12, 0.16], a 1×2 row vector.
- 4Verify: A⁺A = 0.12 × 3 + 0.16 × 4 = 0.36 + 0.64 = 1.
Pseudoinverse of a unit basis vector
When one entry is zero, the pseudoinverse behaves like a simple reciprocal.
- 1Compute the squared norm: 1² + 0² = 1.
- 2Divide each entry by 1: 1/1 = 1, 0/1 = 0.
- 3Result: A⁺ = [1, 0].
Pseudoinverse with a negative entry
Signs carry through the division just like any other scalar operation.
- 1Compute the squared norm: (-6)² + 8² = 36 + 64 = 100.
- 2Divide each entry by 100: -6/100 = -0.06, 8/100 = 0.08.
- 3Result: A⁺ = [-0.06, 0.08].
Introduction
The Pseudoinverse Calculator computes the Moore-Penrose pseudoinverse of a full-rank 2×1 column vector, a fundamental concept in linear algebra used to solve systems that don't have a standard matrix inverse. Because a rectangular matrix can never have a true two-sided inverse, the pseudoinverse provides the best possible generalization, and this tool walks through the exact formula, verifies the identity property, and connects the math to its real use in least-squares regression and data fitting.
What is a pseudoinverse?
The Moore-Penrose pseudoinverse generalizes the matrix inverse to matrices that are not square, or that are square but singular. For a full column-rank matrix A, the pseudoinverse is A⁺ = (AᵀA)⁻¹Aᵀ, and it satisfies A⁺A = I (the identity) on the smaller dimension.
Works for matrices that don't have a regular inverse.
Reduces to the regular inverse when A is square and invertible.
Satisfies A⁺A = I for full column-rank matrices.
How the formula works for a column vector
For a 2×1 column vector A = [a1; a2], the product AᵀA collapses to a single scalar: a1² + a2². Since inverting a scalar just means dividing, the pseudoinverse becomes A⁺ = Aᵀ / (a1² + a2²), a simple 1×2 row vector.
AᵀA is a 1×1 scalar equal to the squared vector norm.
Inverting a scalar is just taking its reciprocal.
The result is a 1×2 row vector, not a 2×2 matrix.
Using the inputs correctly
Enter the two entries of the column vector, a1 and a2. They cannot both be zero, since that would make the squared norm zero and the pseudoinverse undefined (division by zero).
a1 and a2 can be any finite numbers, including negatives and decimals.
At least one entry must be non-zero.
This calculator assumes a full-rank (non-zero) vector.
Reading the outputs
The two pseudoinverse entries form the 1×2 result. The squared norm shows the scalar that was inverted, and the identity check confirms that multiplying the pseudoinverse back by the original vector produces 1, as the theory predicts.
- pinv1 / pinv2:
the two entries of A⁺.
- normSquared:
the value of AᵀA used in the division.
- identityCheck:
A⁺A, which should equal 1 for a valid full-rank vector.
A dependable step-by-step workflow
These four steps mirror exactly how the calculator processes the vector.
Form the column vector from a1 and a2.
Compute AᵀA, the squared norm.
Take the reciprocal of that scalar.
Multiply by Aᵀ to get the pseudoinverse row vector.
Common mistakes and how to avoid them
A common error is confusing the pseudoinverse with a regular matrix inverse, which does not exist for non-square matrices. Another is forgetting that both entries cannot be zero, which would make the norm zero and the division undefined.
Don't expect a square pseudoinverse for a rectangular matrix.
Never allow both entries to be zero simultaneously.
Remember the result is a row vector, not a column vector.
Where the pseudoinverse shows up
The pseudoinverse is central to least-squares regression, where it provides the best-fit solution to overdetermined systems of equations, and it appears throughout data science, robotics (inverse kinematics), and control theory.
Least-squares regression and curve fitting.
Solving overdetermined or underdetermined linear systems.
Robotics: computing joint movements from desired end-effector positions.
Signal processing and machine learning weight estimation.
Reference patterns to remember
A quick reference table for common column vectors helps you sanity-check your own calculations.
Unit vectors have a pseudoinverse equal to their own transpose.
Scaling a vector by k scales its pseudoinverse by 1/k.
The identity check should always equal 1 for a valid, non-zero vector.
| Vector [a1; a2] | Pseudoinverse | Norm² |
|---|---|---|
| [1; 0] | [1, 0] | 1 |
| [3; 4] | [0.12, 0.16] | 25 |
| [0; 5] | [0, 0.2] | 25 |
Quick Reference Card
Pseudoinverse Cheat Sheet
Quick reference • Pseudoinverse Calculator
A⁺ = Aᵀ / (a1² + a2²)Valid range: Any finite a1, a2 with at least one non-zero
Common Values
⚠ Watch Out
- •Both entries cannot be zero simultaneously.
- •The result is a row vector, not a column vector.
- •This formula only applies to full column-rank vectors/matrices.
- •Check domain restrictions before trusting the final value.
Pro Tips
- →Always check that A⁺A equals 1 to confirm your computation.
- →Remember scaling the vector by k scales the pseudoinverse by 1/k.
- →For larger matrices, use SVD-based methods instead of this direct formula.
- →Estimate the answer mentally first so large errors stand out.
FAQs
How is the pseudoinverse different from a regular inverse?
A regular inverse only exists for square, non-singular matrices. The pseudoinverse generalizes this concept to any matrix, including rectangular ones, by finding the best possible approximation that satisfies key inverse-like properties.
Why can't both a1 and a2 be zero?
If both entries are zero, the squared norm AᵀA also equals zero, and the pseudoinverse formula requires dividing by that value. Division by zero is undefined, so the vector must be non-zero.
What does the identity check confirm?
It verifies that A⁺A equals 1, the expected identity result for a full-rank column vector's pseudoinverse. If this doesn't equal 1, something in the input or calculation is off.
Why is the result a row vector instead of a column vector?
The pseudoinverse of an m×n matrix is always n×m. Since the input is a 2×1 column vector, its pseudoinverse is a 1×2 row vector, matching the transposed dimensions.
Does the pseudoinverse always exist?
For any matrix, the Moore-Penrose pseudoinverse always exists and is unique, even for singular or rectangular matrices, though the exact formula used to compute it can vary based on rank and shape.
How is the pseudoinverse used in regression?
In least-squares regression, the pseudoinverse provides the coefficient vector that minimizes the squared error between predicted and observed values, even when the system of equations has no exact solution.
Can this calculator handle larger matrices?
This calculator focuses on the clean, illustrative case of a 2x1 column vector. Larger or non-vector matrices require more general methods like singular value decomposition (SVD).