如何在 R 中制作对称三对角矩阵（Wilkinson 矩阵）并计算特征值？
How can I make a symmetric tridiagonal matrix (Wilkinson matrix) in R and compute the eigenvalues?
In linear algebra, Wilkinson matrices are symmetric, tridiagonal,
order-N matrices with pairs of nearly, but not exactly, equal
eigenvalues. Wilkinson matrices have applications in many fields, including scientific computing, numerical linear algebra, and signal processing.(Wikipedia)
根据维基百科的矩阵示例，我有这个矩阵：
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 3 1 0 0 0 0 0
[2,] 1 2 1 0 0 0 0
[3,] 0 1 1 1 0 0 0
[4,] 0 0 1 0 1 0 0
[5,] 0 0 0 1 1 1 0
[6,] 0 0 0 0 1 2 1
[7,] 0 0 0 0 0 1 3
如何创建此矩阵并计算 R 中的特征值？
使用此脚本，您可以创建威尔金森矩阵并计算特征值
identity_6_by_7 <- diag(rep.int(1, 6), 6, 7) # create a 6x7 matrix with ones on the main diagonal
below_the_diagonal <- rbind(0, identity_6_by_7) # create a row of zeros below the diagonal
identity_7_by_6 <- diag(rep.int(1, 6), 7, 6) # matrix with the ones offset one up from the diagonal
above_the_diagonal <- cbind(0, identity_7_by_6) # create a row of zeros above the diagonal
on_the_diagonal <- diag(abs(seq.int(-3, 3))) # diagonal of values from abs(-3 to 3)
wilkinson_21 <- below_the_diagonal + above_the_diagonal + on_the_diagonal
eigen(wilkinson_21)$values # eigen values
你可以检查本征：
eigen(wilkinson_21)$values
[1] 3.7615572 3.7320508 2.3633282 2.0000000 1.0000000 0.2679492 -1.1248854
In linear algebra, Wilkinson matrices are symmetric, tridiagonal, order-N matrices with pairs of nearly, but not exactly, equal eigenvalues. Wilkinson matrices have applications in many fields, including scientific computing, numerical linear algebra, and signal processing.(Wikipedia)
根据维基百科的矩阵示例，我有这个矩阵：
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 3 1 0 0 0 0 0
[2,] 1 2 1 0 0 0 0
[3,] 0 1 1 1 0 0 0
[4,] 0 0 1 0 1 0 0
[5,] 0 0 0 1 1 1 0
[6,] 0 0 0 0 1 2 1
[7,] 0 0 0 0 0 1 3
如何创建此矩阵并计算 R 中的特征值？
使用此脚本，您可以创建威尔金森矩阵并计算特征值
identity_6_by_7 <- diag(rep.int(1, 6), 6, 7) # create a 6x7 matrix with ones on the main diagonal
below_the_diagonal <- rbind(0, identity_6_by_7) # create a row of zeros below the diagonal
identity_7_by_6 <- diag(rep.int(1, 6), 7, 6) # matrix with the ones offset one up from the diagonal
above_the_diagonal <- cbind(0, identity_7_by_6) # create a row of zeros above the diagonal
on_the_diagonal <- diag(abs(seq.int(-3, 3))) # diagonal of values from abs(-3 to 3)
wilkinson_21 <- below_the_diagonal + above_the_diagonal + on_the_diagonal
eigen(wilkinson_21)$values # eigen values
你可以检查本征：
eigen(wilkinson_21)$values
[1] 3.7615572 3.7320508 2.3633282 2.0000000 1.0000000 0.2679492 -1.1248854