An idempotent matrix represents a projection transformation. When you apply it once or twice to a vector, you get the same result - the vector has been "projected" onto a subspace.
⚠The only invertible idempotent matrix is the identity matrix (A² = A and invertible implies A = I).
What is an Idempotent Matrix?
An idempotent matrix is a matrix that equals its own square (A² = A). When such a matrix is used as a linear transformation, applying it repeatedly produces the same result as applying it once. This is analogous to projection in geometry.
Eigenvalues 0 or 1
Only eigenvalues possible are 0 and 1.
Trace = Rank
For idempotent matrices, trace(A) = rank(A).
Projection
Projects vectors onto the eigenspace for eigenvalue 1.
Determinant
det = 0 (unless matrix is identity).
Example: P = [[1,0],[0,0]] (projects onto x-axis)
1. P² = [[1,0],[0,0]] · [[1,0],[0,0]] = [[1,0],[0,0]] = P ✓
2. P is idempotent ✓
3. For any (x,y): P(x,y) = (x,0)
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.