3.119 min read

Bijectivity and Square Matrices

A map is bijective if it is both injective and surjective — every output is reached, and each output has exactly one pre-image. Bijections have inverses.

For $A: \mathbb{R}^n \to \mathbb{R}^m$ to be bijective, we need both $m = \text{Rank}(A)$ (surjective) and $\text{Nullity}(A) = 0$ (injective). By the Rank-Nullity theorem (Section 3.17), this forces $m = n$ : square matrices.

So: only square matrices can be bijective. And for a square $n \times n$ matrix, surjectivity $\Leftrightarrow$ injectivity $\Leftrightarrow$ bijectivity. You only need to check one.

Formal View

Theorem 3.11 (Collapse Theorem)

For a square matrix

A \in \mathbb{R}^{n \times n}

A \text{ injective} \Longleftrightarrow A \text{ surjective} \Longleftrightarrow A \text{ bijective}

This equivalence fails for non-square matrices.

Why This Matters

The Collapse Theorem is what makes square matrices special — one condition implies all.

For square linear systems: unique solution iff the coefficient matrix is bijective iff it is invertible
In cryptography, bijective linear maps over finite fields are used in block ciphers

Learning Resources

Bijective functions and inverses

Khan Academy

Bijections, bijective functions, and invertibility.

10 min

Quiz

Question 1

A $3 \times 3$ matrix with rank 2 is bijective.

Common Mistakes

Applying the Collapse Theorem to non-square matrices — it ONLY holds for square matrices.
Thinking bijectivity is harder to check than surjectivity or injectivity for square matrices — they are equivalent.