ضرب المصفوفات والمحددات

إنشاء المصفوفات في R

# Create a 3x3 matrix (filled column-wise)
A <- matrix(c(1, 2, 3,
              4, 5, 6,
              7, 8, 9), nrow = 3, ncol = 3, byrow = TRUE)
print(A)
#      [,1] [,2] [,3]
# [1,]    1    2    3
# [2,]    4    5    6
# [3,]    7    8    9

dim(A)   # 3 3
nrow(A)  # 3
ncol(A)  # 3

ضرب المصفوفات: %*%

A <- matrix(c(1, 2, 3, 4), nrow = 2)
B <- matrix(c(5, 6, 7, 8), nrow = 2)

# Element-wise multiplication (NOT matrix mult)
A * B
#      [,1] [,2]
# [1,]    5   21
# [2,]   12   32

# True matrix multiplication
A %*% B
#      [,1] [,2]
# [1,]   19   43
# [2,]   22   50

# Verify: entry [1,1] = 1*5 + 2*6 = 17... wait:
# A[1,] = c(1,3), B[,1] = c(5,6): 1*5 + 3*6 = 23

فهم نتائج %*%

A <- matrix(c(1, 2, 3, 4, 5, 6), nrow = 2, byrow = TRUE)
B <- matrix(c(7, 8, 9, 10, 11, 12), nrow = 3, byrow = TRUE)
# A is 2x3, B is 3x2 -> result is 2x2
C <- A %*% B
print(C)

# Verify entry C[1,1]:
# row 1 of A: c(1,2,3), col 1 of B: c(7,9,11)
manual_11 <- sum(A[1, ] * B[, 1])
manual_11  # 1*7 + 2*9 + 3*11 = 58
C[1, 1]   # should match

المحدِّد: det()

# 2x2 matrix: det = a*d - b*c
A <- matrix(c(3, 2, 1, 4), nrow = 2)
det(A)  # 3*4 - 2*1 = 10

# Singular matrix: det = 0
S <- matrix(c(1, 2, 2, 4), nrow = 2)
det(S)  # 0 -> not invertible

# Larger matrix
B <- matrix(c(2, -1, 0,
             -1,  2, -1,
              0, -1,  2), nrow = 3, byrow = TRUE)
det(B)  # 4

الأثر: sum(diag())

A <- matrix(c(4, 2, 1,
              3, 5, 0,
              2, 1, 6), nrow = 3, byrow = TRUE)
# Extract diagonal elements
diag(A)  # c(4, 5, 6)

# Trace = sum of diagonal
trace_A <- sum(diag(A))
trace_A  # 15

# Trace = sum of eigenvalues (verify)
eigs <- eigen(A)$values
sum(Re(eigs))  # should also be approximately 15

# trace() in base R is NOT the matrix trace:
# trace(A) # this does something else

crossprod() وtcrossprod()

A <- matrix(c(1, 2, 3, 4, 5, 6), nrow = 3)
B <- matrix(c(7, 8, 9, 10, 11, 12), nrow = 3)

# t(A) %*% B  (the slow way)
result1 <- t(A) %*% B

# crossprod(A, B) = t(A) %*% B (faster)
result2 <- crossprod(A, B)
all.equal(result1, result2)  # TRUE

# crossprod(A) = t(A) %*% A
ATA <- crossprod(A)
print(ATA)

# tcrossprod(A, B) = A %*% t(B)
result3 <- tcrossprod(A, B)
all.equal(A %*% t(B), result3)  # TRUE

مصفوفة الوحدة باستخدام diag()

# 3x3 identity matrix
I3 <- diag(3)
print(I3)
#      [,1] [,2] [,3]
# [1,]    1    0    0
# [2,]    0    1    0
# [3,]    0    0    1

# A * I = A (identity property)
A <- matrix(c(2, 3, 4, 5), nrow = 2)
all.equal(A %*% diag(2), A)  # TRUE

# Create diagonal matrix from vector
D <- diag(c(2, 5, 10))
print(D)
det(D)  # Product of diagonal: 100

قوة المصفوفة والضرب المتكرر

# Matrix square
A <- matrix(c(1, 1, 1, 0), nrow = 2)
A_sq <- A %*% A
print(A_sq)
# Fibonacci connection: A^n gives Fib numbers!

# Simple matrix power function
mat_pow <- function(M, n) {
  result <- diag(nrow(M))  # start with identity
  for (i in seq_len(n)) result <- result %*% M
  result
}

mat_pow(A, 5)  # A to the 5th power
# [1,]  8  5
# [2,]  5  3

الجداء الخارجي باستخدام %o%

x <- c(1, 2, 3)
y <- c(10, 20)

# Outer product: 3x2 matrix
P <- x %o% y
print(P)
#      [,1] [,2]
# [1,]   10   20
# [2,]   20   40
# [3,]   30   60

# Same as outer(x, y)
identical(P, outer(x, y))  # TRUE

# Outer with custom function
outer(1:3, 1:3, FUN = '+')
#      [,1] [,2] [,3]
# [1,]    2    3    4
# [2,]    3    4    5
# [3,]    4    5    6

خصائص المحدِّد

A <- matrix(c(2, 1, 5, 3), nrow = 2)
B <- matrix(c(4, 2, 1, 6), nrow = 2)

# det(A %*% B) = det(A) * det(B)
det(A %*% B)
det(A) * det(B)

# det(t(A)) = det(A)
det(t(A)); det(A)

# det(2A) = 2^n * det(A) for n=2
det(2 * A); 2^2 * det(A)

# det of upper triangular = product of diagonal
U <- matrix(c(3, 0, 0, 4, 2, 0, 1, 5, 7), nrow = 3, byrow = TRUE)
det(U); prod(diag(U))  # both = 42

تطبيق عملي: المعادلات الطبيعية

# Generate simple regression data
set.seed(42)
n <- 20
x <- runif(n, 0, 10)
y <- 2 + 3 * x + rnorm(n)

# Design matrix with intercept column
X <- cbind(1, x)

# Check det of X'X (should be nonzero)
XtX <- crossprod(X)  # t(X) %*% X
det(XtX)  # large positive value

# Solve normal equations: beta = solve(X'X) %*% X'y
Xty <- crossprod(X, y)  # t(X) %*% y
beta <- solve(XtX) %*% Xty
beta  # Should be close to c(2, 3)

# Compare with lm()
coef(lm(y ~ x))  # same result

تحقّق سريع

مراجعة: عمليات المصفوفات

A <- matrix(c(2, 1, 1, 3), nrow = 2)
B <- matrix(c(1, 0, 0, 1), nrow = 2)  # Identity

# Core matrix operations summary:
A %*% B             # matrix multiply -> A
det(A)              # 2*3 - 1*1 = 5
sum(diag(A))        # trace = 2+3 = 5
crossprod(A)        # t(A) %*% A
tcrossprod(A)       # A %*% t(A)
diag(3)             # 3x3 identity
diag(c(1,2,3))      # diagonal matrix
A %o% c(1, 2)       # outer product