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A matrix represents a linear projection when applying it twice has the same effect as applying it once. For an orthogonal projection with the standard Hermitian inner product on ${\mathbb{C}}^m$, the additional condition is that the conjugate transpose equals the original matrix:Thus $P$ is both idempotent and a Hermitian matrix. Over the real numbers this says $P^2=P=P^T$. Its image of a linear map is the retained linear subspace, while its kernel of a linear map is the orthogonal complement: vectors in the image are unchanged, and vectors in the kernel become zero.

For a matrix $A$ with full column rank, projecting onto its column space givesThe invertibility of $A^{†}A$ follows from linear independence of the columns. If those columns are an orthonormal set, $A^{†}A=I$ and $P=A{A}^{†}$. These formulas and the conditions $P^2=P=P^T$ in the real case are developed in MIT 18.06SC, Projections onto Subspaces, pp. 1–3.

For the line spanned by $e=(1,2{)}^T$,In particular, $Pe=e$ and $P(2,-1{)}^T=0$. The eigenvalues are $1$ on the retained direction and $0$ on the removed direction. This exposes the meaning of an orthogonal projection as retaining selected components, rather than stretching them.

Symmetry refers to the specified inner product. In the real weighted setting, the formula becomes $P_M=A(A^{T}MA{)}^{-1}{A}^{T}M$ and satisfies $P_M^2=P_M$ and $P_M^{T}M=M{P}_M$. For the preceding weighted line example,It is not symmetric under the ordinary transpose, but it is an orthogonal projection for the weighted Euclidean inner product. The matrix criterion $P=P^{†}$ should therefore be used with the standard Hermitian inner product, not silently transferred to a different geometry.

An orthogonal projection residual is the difference $r=x-P_{W}x$ between a vector and its orthogonal projection onto a linear subspace $W$. It is orthogonal to every vector in $W$, so $⟨r,w⟩=0$ for all $w\in W$. The Gram-Schmidt process uses such residuals to remove directions already represented by earlier basis vectors.

This is an optimality condition as well as a geometric description. No permitted change within $W$ can remove a further orthogonal component of $r$. In the preceding dot product example, $r=(2,-1{)}^T$ is perpendicular to the line spanned by $(1,2{)}^T$.

The selected inner product matters. Keep $x=(3,1{)}^T$ and the same line, but use the weighted Euclidean inner product with $M=\mathrm{diag}(4,1)$. The projection coefficient and residual becomeNow $4(r_M{)}_1{e}_1+(r_M{)}_2{e}_2=5-5=0$, although $r_M\cdot e=-15/4\ne 0$. The orthogonality is with respect to $M$, so it need not look perpendicular in the ordinary drawing. The squared weighted error isThe weighted optimum reduces the first-coordinate error from $2$ to $5/4$, at the cost of increasing the magnitude of the second-coordinate error from $1$ to $5/2$. The weights express which errors are more costly.

In linear least squares, residual orthogonality to the column space is expressed by the normal equations for linear least squares. With a weighted Euclidean inner product, the corresponding real-matrix equation is $A^{T}M(x-Ac)=0$. This makes the choice of error measurement part of the mathematical model, rather than merely a plotting convention.

For a positive weight $w$, the weighted inner product of functions is $⟨f,g{⟩}_w=\int \overline{f}gw$. It is the usual inner product in the L2 space for the measure $w(x)dx$. A Sturm-Liouville eigenfunction expansion uses this weight to make its eigenfunctions orthogonal.

The weight specifies how different locations contribute to a squared error: $\Vert f-g{\Vert }_w^2=\int |f-g{|}^{2}w$. More weight makes an error in that region more costly. The weighted Euclidean inner product makes the same choice for a finite list of coordinates.

For an explicit real-function example on $[0,1]$, let $w(x)=1+x$ and approximate $f(x)=x$ by a constant function. The orthogonal projection onto a finite-dimensional subspace spanned by the function $1$ givesThe best unweighted constant is $1/2$. Giving greater weight to larger $x$ moves the best constant upward, so a weighting choice changes the approximation even though the allowed family of functions is unchanged.