knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
Seja $A_{a,b,n}$ $\in$ $\mathbf{R}^{n\times n}$ uma matriz tri-diagonal dada por: $$A_{a,b,n}=\begin{bmatrix} a & b & 0 & \cdots & 0 & 0\ b & a & b & \ddots & \ddots & \vdots\ 0 & b & a & b & \ddots & \vdots\ \vdots & \ddots & \ddots & \ddots & \ddots & 0\ \vdots & \ddots & \ddots & b & a & b\ 0 & \cdots & \cdots & 0 & b & a \end{bmatrix} _{n,n}$$
A <- matrix(c(8,2,0,0, 2,8,2,0, 0,2,8,2, 0,0,2,8),nrow=4,ncol=4) print(A) cho <- INF2471.P1::cholesky(A) print(cho) print(cho%*%t(cho))
print('R')
Determine uma base ortonormal para o espaço complementar ortogonal ao vetor $v=[1,-1,1]$ $\in$ $\mathbf{R}^3$
v <- matrix(c(1, -1, 1), byrow = T, nrow = 1) base <- INF2471.P1::find_orthogonal_complement(v) print(base) base_ortonormal <- INF2471.P1::apply_Gram_Schmidt(base) print(base_ortonormal) #Dois vetores são ortonormais entre si, se o produto interno deles #é igual a zero e se possuem norma 1. #Produto Interno print(INF2471.P1::dotprod(base_ortonormal[1, ], base_ortonormal[2, ])) #Norma print(INF2471.P1::magnitude(base_ortonormal[1, ])) print(INF2471.P1::magnitude(base_ortonormal[2, ]))
Calcule a pseudoinversa de A: $$A=\begin{bmatrix} 1 & 0\ 0 & 1\ 1 & 1 \end{bmatrix}$$
A <- matrix(c(1, 0, 1, 0, 1, 1), ncol = 2) print(A) pseudoinverse <- INF2471.P1::apply_pseudo_inverse(A) print(pseudoinverse)
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