Lagmental Vicfred

Gram--Schmidt Produces an Orthonormal Basis by Vicfred

tags: mathematics, inner products, spectral theorem, quadratic forms

Subtracting projections onto earlier vectors turns an independent list into orthogonal directions. A small computation will anchor the general statement before the abstraction takes over.

Notation

An inner product \(\langle x,y\rangle\) converts algebraic decompositions into orthogonal ones. Self-adjoint maps satisfy \(T=T^\ast\) and have real spectral data.

$$ u_k=v_k-\sum_{j<k}\frac{\langle v_k,u_j\rangle}{\langle u_j,u_j\rangle}u_j $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ e_k=\frac{u_k}{\|u_k\|} $$

Stress the formula

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \begin{pmatrix}v_1&\cdots&v_n\end{pmatrix}=QR,\qquad Q^\ast Q=I,\quad R\ \text{upper triangular} $$

Interpretation

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad u_k=v_k-\sum_{j<k}\frac{\langle v_k,u_j\rangle}{\langle u_j,u_j\rangle}u_j,\\[5pt] \mathsf{C}\;&:\quad e_k=\frac{u_k}{\|u_k\|}. \end{aligned} $$

Limit of the argument

Orthogonal diagonalization requires self-adjointness over the real or complex inner-product setting. A general diagonalizable matrix need not have orthogonal eigenvectors.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] e_k=\frac{u_k}{\|u_k\|} \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Thu 11 June 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.