# Tag Info

## New answers tagged orthogonality

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### Substantiate the following statement " Arnoldi iteration is nothing but orthogonal projection onto Krylov subspaces"

Let $A \in \mathbb{R}^{m \times m}$ and let $b \in \mathbb{R}^{m}$. Let $n \leq m$ denote the smallest $n$ such that there exists a monic polynomial $p$ of degree $n$ such that $p(A)b = 0$. It follows ...
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### Weight function given polynomial basis

The claim does not hold. The roots of two consecutive polynomials must interlace. Still this additional condition is not sufficient. Indeed WLOG we may assume that the polynomials are monic, i.e. that ...
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### Condition of orthogonality of two solutions to a 2nd order ODE with constant coefficients on a smooth chart

Let $\tilde{u},\tilde{v}$ be the other solution of the differential equation. Dividing by $\dot{v}^2$, I get $A(\frac{\dot{u}}{\dot{v}})^2+2 B \frac{\dot{u}}{\dot{v}}+C =0,$ so the two solutions of ...
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### Orthogonal projection is bounded

First you need the fact that $P^*=P$. Indeed, $$\langle P^*(u+w),u+w\rangle=\langle u+w,P(u+w)\rangle=\langle u+w,u\rangle=\langle u,u\rangle=\langle u,u+w\rangle=\langle P(u+w),u+w\rangle.$$ It ...
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