Gram-Schmidt process

Japanese school year: University year 2

What you learn

Learn the algorithm to construct an orthonormal basis for the subspace spanned by a set of linearly independent vectors. This method is fundamental for matrix QR decomposition and generating systems of orthogonal functions. Understanding inner products, vector projections, and vector space bases is required.

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Key points

This formula is used to construct a new orthogonal (mutually perpendicular) vector $\boldsymbol{u}_k$ from a given vector $\boldsymbol{v}_k$ . It subtracts the projection components along all previously found orthogonal vectors $\boldsymbol{u}_j$ from $\boldsymbol{v}_k$ .

\boldsymbol{u}_k = \boldsymbol{v}_k - \sum_{j=1}^{k-1} \frac{\langle \boldsymbol{v}_k, \boldsymbol{u}_j \rangle}{\|\boldsymbol{u}_j\|^2} \boldsymbol{u}_j

This formula is used to normalize an orthogonal vector $\boldsymbol{u}_k$ to length 1 without changing its direction. Dividing $\boldsymbol{u}_k$ by its length (norm $\|\boldsymbol{u}_k\|$ ) gives the unit vector $\boldsymbol{e}_k$ .

\boldsymbol{e}_k = \frac{\boldsymbol{u}_k}{\|\boldsymbol{u}_k\|}

This formula is used to find the orthogonal projection vector (perpendicular shadow) of vector $\boldsymbol{v}$ onto vector $\boldsymbol{u}$ . It multiplies vector $\boldsymbol{u}$ by the inner product $\langle \boldsymbol{v}, \boldsymbol{u} \rangle$ divided by the squared length $\|\boldsymbol{u}\|^2$ .

\operatorname{proj}_{\boldsymbol{u}}(\boldsymbol{v}) = \frac{\langle \boldsymbol{v}, \boldsymbol{u} \rangle}{\|\boldsymbol{u}\|^2} \boldsymbol{u}

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Practice 1Coming soon