计算矩阵的逆矩阵:系统是计算奇异的[错误]

Question

我有一个矩阵m:

(m <- matrix(c(26,14,24,14,20,42,24,42,90), 3))

#      [,1] [,2] [,3]
# [1,]   26   14   24
# [2,]   14   20   42
# [3,]   24   42   90

当我运行solve(m)计算矩阵的逆时,我收到此错误消息:

solve(m)

solve.default(m)出错:系统计算奇异:倒数条件数= 6.21104e-18

Answer 1

我们可以看到,必须以多种方式实现这一点,每种方式都意味着不可逆性:

1)行列式m为零:

> det(m)
[1] -2.685852e-12

2) m具有零特征值,即eigen(m)$values[3].等效地,nullspace m是非null - 它等于跨越的1维空间eigen(m)$vectors[, 3]

> e <- eigen(m); e
$values
[1]  1.180000e+02  1.800000e+01 -6.446353e-15

$vectors
           [,1]          [,2]       [,3]
[1,] -0.2881854  9.486833e-01  0.1301889
[2,] -0.4116935  1.110223e-16 -0.9113224
[3,] -0.8645563 -3.162278e-01  0.3905667

> N <- e$vector[, 3]  # nullspace
> m %*% N  # shows that N is indeed mapped to zero
             [,1]
[1,] 5.329071e-15
[2,] 0.000000e+00
[3,] 0.000000e+00

3)的列m是不线性独立的.特别是m[,1]在其他列上的回归给出了完美的拟合(即拟合值相等m[, 1]),因此从线性模型的系数得到我们的 7 * m[,2] - 3 * m[, 3]等于m[, 1].

> fm <- lm(m[, 1] ~ m[, 2] + m[, 3] + 0)

> all.equal(fitted(fm), m[, 1]) # perfect fit
[1] TRUE

> coef(fm)
m[, 2] m[, 3] 
     7     -3 

> all.equal(7 * m[, 2] - 3 * m[, 3], m[, 1])
[1] TRUE

4) cholesky分解在其对角线上为零:

> chol(m, pivot = TRUE)
         [,1]     [,2]      [,3]
[1,] 9.486833 2.529822 4.4271887
[2,] 0.000000 4.427189 0.6324555
[3,] 0.000000 0.000000 0.0000000
attr(,"pivot")
[1] 3 1 2
attr(,"rank")
[1] 2
Warning message:
In chol.default(m, pivot = TRUE) :
  the matrix is either rank-deficient or indefinite

5) m不是满级,即等级小于3:

> attr(chol(m, pivot = TRUE), "rank")
[1] 2
Warning message:
In chol.default(m, pivot = TRUE) :
  the matrix is either rank-deficient or indefinite

注意:输入可通过以下方式重复给出:

m <- matrix(c(26, 14, 24, 14, 20, 42, 24, 42, 90), 3)

Answer 2

问题是列不是线性独立的.

第一列*-1/3 +第二列*7/3等于第三列.

-m[, 1] * 1/3 + 7/3 * m[, 2]

# [1] 24 42 90