# RPD -- Various Ch. 2 Examples



# Example 2.1

TT <- matrix(c(1,4,3,2,5,0), 3, 2)

  # must use name TT rather than T --- T is reserved for boolean "true"

W <- rbind(-1, 3)

TTW <- TT %*% W           # matrix multiplication

print(TT)
print(W)
print(TTW)

TTtrans <- t(TT)          # transpose

print(TTtrans)



# Example 2.2: Singular Matrix

A <- matrix(c(2,1,5,4,2,7,6,3,9), 3, 3)

detA <- det(A)            # determinant

print(A)
print(detA)



# Example 2.3: Matrix Inverse

B <- matrix(c(1,4,8,3,5,7,2,6,9), 3, 3)

detB <- det(B)

invB <- solve(B)          # inverse

print(B)
print(detB)
print(invB)



# Example 2.4: Matrix Rank

A <- matrix(c(1,2,3,2,4,3,3,6,3), 3, 3)

rankA <- qr(A)$rank       # rank, based on a numerical decomposition

print(A)
print(rankA)

Y <- rbind(6, 10, 9)

A.Y <- cbind(A, Y)        # combine columns into one matrix

rankA.Y <- qr(A.Y)$rank

print(A.Y)
print(rankA.Y)

  # Note:  There is an R function named "rank", but it does something else.



# Example 2.10: Projections

A <- (1/6) * matrix(c(5,2,-1,2,2,2,-1,2,5), 3, 3)

AA <- A %*% A

traceA <- sum(diag(A))    # trace

print(A)
print(AA)
print(traceA)

ImA <- diag(1,3) - A      # diag(1,n) = n-by-n identity

ImAImA <- ImA %*% ImA

traceImA <- sum(diag(ImA))

print(ImA)
print(ImAImA)
print(traceImA)

AImA <- A %*% ImA         # zero, except for numerical error

print(AImA)