Why QDU solves the projected least squares system

Key takeaway: For an inconsistent linear system, the factorization A = QDU helps separate the column-space geometry from the algebra. The equation (QDU)X = B is not really being solved against the original B in the full space; it is solved after projecting B onto col(A), which is the part A can actually produce.

The key idea: only vectors in col(A) are attainable

If the system $AX=B$ is inconsistent, that means $B$ is not in the column space of $A$. So there is no vector $X$ such that $AX$ equals the original $B$ exactly. What the factorization $A=QDU$ gives you is a clean way to separate the part of $B$ that lies inside $\operatorname{col}(A)$ from the part that lies outside it.

Because the columns of $A$ span $\operatorname{col}(A)$, any exact output of $AX$ must live in that subspace. If $B$ has a component orthogonal to $\operatorname{col}(A)$, that component cannot be matched by any choice of $X$.

Why solving $(QDU)X=B$ really means solving against the projection

In practice, the $Q$ factor is orthogonal, so it preserves lengths and angles. The decomposition is arranged so that the action of $QDU$ maps coordinates in the domain into the subspace where the data live. When $B$ is not in $\operatorname{col}(A)$, the best you can do is replace $B$ by its orthogonal projection onto $\operatorname{col}(A)$:

$AX = \operatorname{Proj}_{\operatorname{col}(A)}(B).$

That projected vector is the unique vector in $\operatorname{col}(A)$ closest to $B$. Since every vector of the form $AX$ must lie in $\operatorname{col}(A)$, this is the meaningful target. The factorization lets you solve the reduced, compatible system inside the subspace where a solution actually exists.

Why this is not a true solution to the original system

A true solution to $AX=B$ would require exact equality with $B$, not just the closest vector in the column space. If $B\notin \operatorname{col}(A)$, then the residual

$r = B - AX$

cannot be zero. Instead, for the least-squares or projected solution, that residual is orthogonal to $\operatorname{col}(A)$, which is the geometric condition characterizing the best approximation.

So $(QDU)X=B$ is only a formal way of writing the computation after you reinterpret the right-hand side as its projection. It does not contradict inconsistency, because the actual solved equation is

$AX = \operatorname{Proj}_{\operatorname{col}(A)}(B),$

not $AX=B$.

The geometry behind the factorization

The factorization $A=QDU$ is useful because it isolates three roles: $U$ changes coordinates, $D$ scales along independent directions, and $Q$ rotates or reflects into the subspace geometry. This makes it easier to see that the system is solvable exactly only for right-hand sides inside the image of $A$.

When $B$ is outside that image, the orthogonal projection is the best possible replacement. That is why the factorization naturally leads to the projected system and to least-squares behavior.

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Pitfalls the pros know 👇 A common mistake is to think that writing $(QDU)X=B$ automatically means you have solved the original inconsistent system. That is not correct unless $B$ already lies in $\operatorname{col}(A)$. Another pitfall is to confuse the algebraic manipulation with the geometric meaning: the factorization does not create a true solution out of nothing. It only makes the closest attainable right-hand side explicit. The residual does not disappear; it becomes the part of $B$ orthogonal to the column space.

What if the problem changes? If the system were consistent, meaning $B\in\operatorname{col}(A)$, then the projection would satisfy $\operatorname{Proj}_{\operatorname{col}(A)}(B)=B$, and solving the factored system would give a true exact solution to $AX=B$. A different variant is when you solve the normal equations $A^TAX=A^TB$ instead. That formulation also produces the least-squares solution, but it uses the orthogonality condition directly rather than the $QDU$ geometry. In both cases, the computed $X$ minimizes the error when exact equality is impossible.

Tags: column space projection, least squares solution, orthogonal residual