矩阵语言

(Vinod 2011): Hands-on matrix algebra using R: active and motivated learning with applications.

(Fieller 2015): Basics of Matrix Algebra for Statistics with R.

1 基础矩阵

1.1 记号表达

(Fieller 2015, sec 1.3)

矩阵（matrix）：以加粗大写英文字母表达，例如$\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C}, \boldsymbol{D}, \boldsymbol{U}, \boldsymbol{V}, \boldsymbol{W}, \boldsymbol{X}, \boldsymbol{Y}, \boldsymbol{Z}$.
列向量（column vectors）：以加粗小写英文字母表达，例如$\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}, \boldsymbol{d}, \boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$.
向量的元素（elements）：一般用小写英文字母（单下标）表达，例如列向量$\boldsymbol{x}$的元素记为$x_1, x_2, \ldots, x_p$.
矩阵的元素（elements）：一般用小写英文字母（双下标）表达，例如矩阵$\boldsymbol{X}$的元素记为$x_{11}, x_{12}, \ldots, x_{1 n}, x_{21}, \ldots, \ldots, x_{p n}$.
矩阵的列（columns）：一般用加粗小写英文字母（单下标）表达，例如矩阵$\boldsymbol{X}$的列分别记为$\boldsymbol{x}_1, \boldsymbol{x}_2, \ldots, \boldsymbol{x}_n$.
整数（integers）：英文字母表的一些中间字母一般用作整数，例如$i, j, k, l, m, n, p, q, r, s, t$
- $i, j, k, l$一般用作虚拟整数或索引整数（dummy or indexing integers）
- $m, n, p, q, r, s, t$则往往用作固定整数（fixed integers）
- 例子：$i=1,2, \ldots, n$；或者$\sum_{i = 1}^{n}{X_i}$

1.2 矩阵运算

R函数的矩阵运算及基本介绍：

A * B: Element-wise multiplication.
A %*% B: Matrix multiplication.
A %o% B: Outer product. $AB'$
crossprod(A,B), crossprod(A): $A'B$ and $A'A$ respectively.
t(A): Transpose
diag(x): Creates diagonal matrix with elements of x in the principal diagonal
diag(A): Returns a vector containing the elements of the principal diagonal
diag(k): If $k$ is a scalar, this creates a $k \times k$ identity matrix. Go figure.
solve(A, b): Returns vector $\text{x}$ in the equation $b = A\text{x} (i.e., A^{-1}b)$
solve(A): Inverse of A where A is a square matrix.
ginv(A): Moore-Penrose Generalized Inverse of A.
ginv(A): requires loading the MASS package.
y <- eigen(A): y$val are the eigenvalues of A
y$vec: are the eigenvectors of A
y <- svd(A): Single value decomposition of A.
- y$d = vector containing the singular values of A
- y$u = matrix with columns contain the left singular vectors of A
- y$v = matrix with columns contain the right singular vectors of A
R <- chol(A): Choleski factorization of A. Returns the upper triangular factor, such that R’R = A.
y <- qr(A): QR decomposition of A.
- y$qr has an upper triangle that contains the decomposition and a lower triangle that contains information on the Q decomposition.
- y$rank is the rank of A.
- y$qraux a vector which contains additional information on Q.
- y$pivot contains information on the pivoting strategy used.
cbind(A,B,...): Combine matrices(vectors) horizontally. Returns a matrix.
rbind(A,B,...): Combine matrices(vectors) vertically. Returns a matrix.
rowMeans(A): Returns vector of row means.
rowSums(A): Returns vector of row sums.
colMeans(A): Returns vector of column means.
colSums(A): Returns vector of column sums.

2 实操杂音

2.1 数值近似问题

```{r}
x1 <- c(14,-34)
x2 <- c(-4, 24)
X <- cbind(x1,x2)
# inverse X
invX <- solve(X)

X
```

      x1 x2
[1,]  14 -4
[2,] -34 24

矩阵乘积$\boldsymbol{X X'=I}$，但实际计算时可能会有浮点值差异（见左边），把精确度稍微减低则可以得到单位矩阵（见右边）。

```{r}
X %*% invX
```

     [,1]          [,2]
[1,]    1 -1.110223e-16
[2,]    0  1.000000e+00

```{r}
round(X %*% invX, 3)
```

     [,1] [,2]
[1,]    1    0
[2,]    0    1

参考

Fieller, Nick. 2015. Basics of Matrix Algebra for Statistics with R. 1 本. Chapman and Hall/CRC. https://doi.org/10.1201/b18669.

Vinod, Hrishikesh D. 2011. Hands-on Matrix Algebra Using R: Active and Motivated Learning with Applications. Singapore: World Scientific.