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Projection Matrix Calculator

Compute orthogonal projection matrix and project vectors

Enter Direction Vector for Line in 2D

Vector to Project (Optional)

Projection Matrix Formula

P = A(A^T A)^{-1} A^T

P² = P (idempotent)

P^T = P (symmetric)

proj_v(x) = Px

The orthogonal projection matrix P projects vectors onto the column space of matrix A. The projection is idempotent (applying it twice gives the same result) and symmetric. The matrix I-P projects onto the orthogonal complement.

⚠ For projection onto a line spanned by vector v, the formula simplifies to P = (vv^T)/(v^T v). Verify: P² = P and Pv = v.

What is a Projection Matrix?

A projection matrix is a square matrix P that is idempotent (P² = P) and symmetric (P^T = P) for orthogonal projections. Projection matrices find applications in least squares, where they project onto the column space of a matrix to minimize squared error, and in many geometric transformations.

Idempotent

P² = P. Applying projection twice is same as once. Already on subspace stays.

Symmetric

P^T = P for orthogonal projection. Ensures minimum distance property.

Orthogonal

x - Px is orthogonal to subspace. Minimizes ||x - Px|| (least squares).

Eigenvalues

Eigenvalues are 0 and 1 only. 1 for subspace, 0 for orthogonal complement.

Teaching Example: Project [2,3] onto line [1,1].
1. v^T v = 1*1 + 1*1 = 2
2. vv^T = [[1,1],[1,1]]
3. P = [[1,1],[1,1]]/2 = [[0.5,0.5],[0.5,0.5]]
4. proj = Px = [(2+3)/2, (2+3)/2] = [2.5,2.5].

Applications

Least Squares Linear Regression Orthogonal Decomp Optimization Computer Graphics

Frequently Asked Questions

What is a projection matrix?▼

P = A(A^T A)^-1 A^T. Projects orthogonally onto column space of A. P² = P.

Projection properties?▼

P²=P (idempotent), P^T=P (symmetric), I-P projects onto orthogonal complement.

Projection onto a line?▼

For vector v: P = (vv^T)/(v^T v). For R²: [[a²,ab],[ab,b²]]/(a²+b²).

Applications of projection?▼

Least squares, linear regression, orthogonal decomposition, optimization, computer graphics.

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