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Let $A_i$ be real symmetric matrices with sum $I_n$. Their matrix ranks sum to at least $n$, because their ranges span ${\mathbb{R}}^n$. If the ranks sum to $n$, those ranges form a direct sum. For $y$ in the range of $A_i$, the identity $y={\sum }_j{A}_{j}y$ and uniqueness of its range decomposition imply $A_{j}y={\delta }_{ij}y$. Thus $A_i^2=A_i$ and $A_i{A}_j=0$ for $i\ne j$: the matrices are mutually orthogonal orthogonal projections. In a combined orthonormal basis, the forms $Z^T{A}_{i}Z$ for $Z\sim N_n(0,I_n)$ use disjoint sets of independent standard normal coordinates. They are independent chi-squared distributions with degrees of freedom equal to the ranks. Conversely, if all these forms have those chi-squared laws, their expectations and their sum give ${\sum }_i\mathrm{rank}{A}_i=n$.

In a fixed full-rank normal linear model, the studentized estimator of one coefficient has a noncentral t-distribution with noncentrality equal to that coefficient divided by its standard deviation. The fitted and residual orthogonal projections of the Gaussian noise are independent. Therefore the statistical power of its exact Student's t-test does not depend on the other regression coefficients, with design and variance fixed.

For a unital star-representation $\pi :C(K)\to \mathcal{B}(H)$, the Riesz-Markov-Kakutani representation theorem gives regular measures ${\mu }_{x,y}$ representing $f\mapsto ⟨\pi (f)x,y⟩$. Define $P(E)$ by $⟨P(E)x,y⟩={\mu }_{x,y}(E)$. Positivity and ${\mu }_{x,x}(K)=\Vert x{\Vert }^2$ make these positive contractions. The identities ${\mu }_{\pi (f)x,y}=f{\mu }_{x,y}$ imply commutation with $\pi (f)$ and ${\mu }_{P(E)x,y}=1_E{\mu }_{x,y}$; hence $P(F)P(E)=P(F\cap E)$. Thus each $P(E)$ is an orthogonal projection. Scalar countable additivity and orthogonality give countable additivity in the strong operator topology, constructing the normalized regular projection-valued measure.

For a finite-dimensional Hermitian matrix $A$, its spectral projection $P_{\lambda }$ is the orthogonal projection onto the eigenspace for $\lambda$. In an orthonormal eigenbasis, the displayed sum is over every basis vector with that eigenvalue. The projection is independent of the choice of basis within a repeated eigenspace. For distinct eigenvalues $\lambda ,\mu$,This is the spectral decomposition grouped by distinct eigenvalues: decompose a vector into orthogonal pieces, then scale each piece. For the matrix $A=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$,Thus $P_3(1,0{)}^T=(1/2,1/2{)}^T$ and $P_1(1,0{)}^T=(1/2,-1/2{)}^T$: the two pieces add back to $(1,0{)}^T$. Squaring the matrix simply squares each scalar, $A^2=9P_3+P_1$, and similarly the matrix exponential is $e^{tA}=e^{3t}{P}_3+e^t{P}_1$. Spectral projections of a self-adjoint matrix turn repeated matrix calculations into scalar calculations on fixed vector subspaces.

The statistical tangent space is the closed linear span in L2 space of a statistical tangent set. Taking this closure makes orthogonal projection available and ensures that a continuous derivative specified on attainable score functions extends to the whole space.

The tangential component is the orthogonal projection of a vector onto the tangent space. For a plane curve, it is $(w\cdot T)T$; for a hypersurface, it is $w-(w\cdot n)n$. The scalar tangential component of acceleration is $\dot{v}$, which can be positive, zero or negative as the speed increases, remains constant or decreases.

Let nonnegative partition-normalized B-splines $N_i$ form a basis, with ${\sum }_i{N}_i\le 1$ on the integration interval and $d_i=\int N_i>0$. The row-normalized mixed Gram matrix is $G_{ij}=\int N_i{N}_j/d_i$. The normal equations for the orthogonal projection onto their linear span become $Ga=b$, with $b_i=\int f{N}_i/d_i$ and $\Vert b{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$. The subpartition of unity for B-splines gives $\Vert {\sum }_i{a}_i{N}_i{\Vert }_{\infty }\le \Vert a{\Vert }_{{\ell }^{\infty }}$ and proves the bound. The ordinary symmetric Gram matrix omits this essential row normalization. The projection maps continuous functions into the spline space, not onto all continuous functions.

For Kraus operators satisfying ${\sum }_i{A}_i^{†}{A}_i=I$, the map $V|\varphi ⟩={\sum }_i{A}_i|\varphi ⟩\otimes |i⟩$ is an isometry. Implement it by a unitary operator on the system and a quantum ancilla, initially in an extra unused state $|0⟩$. Projecting the output quantum ancilla onto $|i⟩$ gives joint unnormalized state $A_i\rho A_i^{†}\otimes |i⟩⟨i|$. The outcome probability is $\mathrm{Tr}(A_i\rho A_i^{†})$, and the conditional system state is $A_i\rho A_i^{†}$ divided by this probability. The unused level can be merged into the first orthogonal projection without changing any outcome on the prepared state, giving a complete projective measurement with exactly the original outcome labels.

// Target: quantum-theory.bigb

The vector projection multiplies the scalar projection by the chosen unit vector, giving the orthogonal projection onto the linear span of a nonzero real Euclidean vector $\mathbf{c}$:For $\mathbf{a}=(3,4)$ and $\mathbf{c}=(2,0)$, this gives $\mathbf{p}=(6/4)(2,0)=(3,0)$ and $\mathbf{r}=(0,4)$; the dot product $\mathbf{r}\cdot \mathbf{c}=0$ checks the orthogonal projection residual. Replacing $\mathbf{c}$ by $(-2,0)$ gives the same $\mathbf{p}=(-6/4)(-2,0)=(3,0)$: reversing direction changes the scalar projection, but preserves the vector projection onto the line.

For a short proof of the nearest-point property, any vector $\mathbf{z}$ on this line satisfies $\Vert \mathbf{a}-\mathbf{z}{\Vert }^2=\Vert \mathbf{r}{\Vert }^2+\Vert \mathbf{p}-\mathbf{z}{\Vert }^2$ by the Pythagorean identity. The second term is uniquely minimized at $\mathbf{z}=\mathbf{p}$. This connects a directional measurement to the orthogonal projection used in approximation.^[ref]

For a unitary operator $U$ on a Hilbert space, the Cesàro averagesconverge strongly to the orthogonal projection onto the fixed-point subspace $\ker (U-I)$.

The wavelet projection is the orthogonal projection onto the approximation space $V_J$. Iterating $V_{j+1}=V_j\oplus W_j$ expresses it as the coarse part of the wavelet series plus details at levels below $J$. Orthogonality gives $\Vert m-P_{J}m{\Vert }_2^2={\sum }_{j\ge J,k}|b_{j,k}{|}^2$, which tends to zero by the Parseval identity for a Hilbertian basis. The projection can also be represented by $A_J(x,y)={\sum }_k{\varphi }_{J,k}(x){\varphi }_{J,k}(y)$ as an integral kernel in the Hilbert-space sense.

For $M>0$ with integral one, the Cauchy-Schwarz inequality makes the mass functional bounded on the weighted Hilbert space. The vector $M$ has norm one, so the velocity-reset projection is the orthogonal projection onto its span. Consequently $L=\Pi -I$ is bounded and self-adjoint. This Hilbert geometry differs from the norm-one positive reset projection on phase-space $L^1$.