amv11 - Matrix Algebra: Spectral Decomposition, Square Root Matrix, and Singular Value Decomposition

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[statisticsmatt]

begin of video 1
# Parentheses around the assignment makes it print.
(A<-matrix(c(23, 29, 18, 3.54, 3.81, 2.75), ncol=2, byrow = FALSE))
A[1, 2] # 1st row, 2nd column
A[3, 1] # 3rd row, 1st column
# R prints vectors as a row vector to save space.
# However, think of this as a column vector.
(a <- c(98, 86, 93, 97))
a[1] # 1st element of vector a
a[4] # 4th element of vector a
plot(2, 3, xlim=c(0, 2), ylim=c(0, 3), xlab="1st Component of Vector", ylab="2nd Component of Vector",
main="Plot of 2D vector (2, 3)")
arrows(0, 0, 2, 3)
k <- 22
print(k)
(A<-matrix(c(3, 1, -2, 3, 4, 7), ncol=3))
(B<-t(A))
(C<-matrix(c(3, 1, -2, 3, 4, 7), ncol=3))
(D<-matrix(c(3, 1, -2, 3, 4, 6), ncol=3))
# Comparing matrices
A==B
A==C
A==D
# Comparing vectors
A[1, ]==D[1, ]
A[2, ]==D[2, ]
end of video 1

begin of video 2
(A<-matrix(c(23, 29, 18, 3.54, 3.81, 2.75), ncol=2, byrow=FALSE))
t(A)
t(t(A))
5 # scalar
t(5)
(A<-matrix(c(3, -2, 4, -2, 10, -7, 4, -7, 9), ncol=3))
all.equal(A, t(A))
diag(c(3, 10, -7))
A
diag(A)
diag(diag(A))
diag(3)
(j<-rep(1, 10))
matrix(rep(1, 10*10), ncol=10)
end of video 2
begin of video 3
(a<-c(2, 6, -4, 1, -2, 7))
A
sum(a)
sum(A)
sum(A[1, ])+sum(A[2, ])
sum(A[1:2, ])
sum(diag(A))
prod(a)
prod(A)
prod(1:5) # not efficient way to calculate 5!
A
(B<-matrix(sample(1:15, 9), ncol=3))
A+B
B+A
t(A)-t(B)
t(A-B)
(A<-matrix(c(2, 7, 3, 4, 6, 9), ncol=3, byrow=TRUE))
2*A
(A<-matrix(c(2, 1, 3, 4, 6, 5), ncol=3, byrow=TRUE))
(B<-matrix(c(1, 4, 2, 6, 3, 8), ncol=2, byrow=TRUE))
A[1, ]%*%B[, 1]
A[1, ]%*%B[, 2]
A[2, ]%*%B[, 1]
A[2, ]%*%B[, 2]
A %*% B
B %*% A
(A<-matrix(c(2, 1, 3, 4, 6, 5), ncol=3, byrow=TRUE))
(B<-matrix(c(1, 4, 2, 6, 3, 8), ncol=2, byrow=TRUE))
(C<-matrix(c(-1, 4, -2, 6, -3, 8), ncol=2, byrow=TRUE))
A%*%(B+C)
A%*%B + A%*%C
(B+C)%*%A
B%*%A + C%*%A
t(A%*%B)
t(B)%*%t(A)
diag(2)%*%A
A%*%diag(3)
end of video 3

begin of video 4

(A<-matrix(c(2, 1, 3, 4, 6, 5), ncol=3, byrow=TRUE))
(b<-c(1, 2, 3))
A%*%b
b[1]*A[, 1] + b[2]*A[, 2] + b[3]*A[, 3, drop=FALSE]
# 'drop' keeps the structure a matrix, otherwise drops it when possible.
# A vector in matrix form is over parameterized.
(A<-matrix(c(3, -2, 4, 1, 3, 5), ncol=3, byrow=TRUE))
(b<-c(2, -5))
t(b)%*%A
b[1]*A[1, ] + b[2]*A[2, ]
b[1]*A[1, ] + b[2]*A[2, , drop=FALSE]
(a<-c(2, -1, 3, 4, -6, 5))
(b<-c(1, 2, 3, 4, 5, 6))
t(a)%*%b
t(b)%*%a
a%*%t(b)
b%*%t(b) # quick way to make a product table

end of video 4

begin of video 5
(j <- rep(1, 10))
t(j)%*%j
j%*%t(j)

(j <- rep(1, 4))
(a <- sample(-5:10, 4, TRUE))
t(j)%*%a
t(a)%*%j

(j <- rep(1, 4))
(A <- matrix(sample(-5:10, 16, TRUE), ncol=4))
t(j)%*%A # column sums
A%*%j # row sums

(A <- matrix(sample(-5:10, 8, TRUE), ncol=2))
t(A)%*%A
t(A[, 1])%*%A[, 1]
t(A[, 1])%*%A[, 2]
t(A[, 2])%*%A[, 1]
t(A[, 2])%*%A[, 2]
A%*%t(A)
t(A[1, ])%*%A[1, ]
t(A[1, ])%*%A[2, ]
t(A[1, ])%*%A[3, ]
t(A[3, ])%*%A[2, ]

(A <- matrix(sample(-5:10, 8, TRUE), ncol=2))
(D <- diag(1:4))
D%*%A
A
(D <- diag(1:2))
A%*%D

end of video 5



begin of video 6

(A <- matrix(c(2, -3, 4, 1, 4, 8, 8, 2, 1), ncol=3))
(B <- matrix(1:9, ncol=3))
(A11 <- A[1:2, 1:2])
(A12 <- A[1:2, 3, drop=FALSE])
(A21 <- A[3, 1:2, drop=FALSE])
(A22 <- A[3, 3, drop=FALSE])
(B11 <- B[1:2, 1:2])
(B12 <- B[1:2, 3, drop=FALSE])
(B21 <- B[3, 1:2, drop=FALSE])
(B22 <- B[3, 3, drop=FALSE])

A%*%B
A11%*%B11 + A12%*%B21
A11%*%B12 + A12%*%B22
A21%*%B11 + A22%*%B21
A21%*%B12 + A22%*%B22

end of video 6


begin of video 7

(A <- matrix(c(3, 1, 2, 6, 2, 4), ncol=2))
(B <- matrix(c(1, 5, -2, 2, 3, 4), ncol=3))
qr(A)$rank # qr is from the QR Decomposition
qr(B)$rank

(A <- matrix(c(2, -3, 4, 1, 4, 8, 8, 2, 1), ncol=3))
(B <- matrix(c(1, 2, 4, 1, -5, 8, 8, 2, 1), ncol=3))
qr(A)$rank # qr is from the QR Decomposition
qr(B)$rank
(A1 <- solve(A))
(B1 <- solve(B))
A1%*%A
zapsmall(A1%*%A)
zapsmall(A%*%A1)
all.equal(solve(solve(A)), A) # property 1
all.equal(solve(t(A)), t(solve(A))) # property 3
all.equal(solve(A%*%B), solve(B)%*%solve(A)) # property 4

end of video 7

begin of video 8

(A <- matrix(c(3, 0, -3, 0, 6, 3, -3, 3, 6), ncol=3))
(T <- chol(A))
sqrt(3)
sqrt(6)
sqrt(1.5)
t(T)%*%T
all.equal(A, t(T)%*%T)

(A <- matrix(c(3, 0, -3, 0, 6, 3, -3, 3, 6), ncol=3))
det(A)
det(A) == det(t(A))
det(solve(A)) == 1/det(A)
det(2*A) == 2^3 * det(A) # example why "==" has problems
all.equal(det(2*A), 2^3 * det(A))
all.equal(det(t(A)%*%A), det(A)^2)

(A <- matrix(c(3, 0, -3, 0, 6, 3, -3, 3, 6), ncol=3))
(B <- matrix(c(1, 2, -3, -1, 6, 3, 4, 3, 1), ncol=3))
sum(diag(A))
all.equal(sum(diag(A+B)), sum(diag(A))+sum(diag(B)))
all.equal(sum(diag(2*A)), 2*sum(diag(A)))
all.equal(sum(diag(A%*%B)), sum(diag(B%*%A)))
@
\LARGE

end of video 8


begin of video 9

(A <- matrix(c(1, -1, 2, 4), ncol=2))
eigen(A)
(A.val <- eigen(A)$values)
(A.vec <- eigen(A)$vectors)
A%*%A.vec[, 1]
A.val[1]*A.vec[, 1, drop=FALSE]
all.equal(A%*%A.vec[, 2], A.val[2]*A.vec[, 2, drop=FALSE])

crossprod(A.vec[, 1]) # dot product with itself
t(A.vec[, 2])%*%A.vec[, 2] # both vectors are normalized
t(A.vec[, 1])%*%A.vec[, 2] # not orthogonal to each other

A <- matrix(c(1, -1, 2, 4), ncol=2)
A.val <- eigen(A)$values
det(A)
prod(A.val)
sum(diag(A))
sum(A.val)

A <- matrix(c(13, -4, 2, -4, 13, -2, 2, -2, 10), ncol=3)
A.val <- eigen(A)$values
A.vec <- eigen(A)$vectors
zapsmall(A.vec %*% t(A.vec))
zapsmall(t(A.vec) %*% A.vec)
eigen(A)$values
qr(A)$rank

end of video 9

begin of video 10

(A <- matrix(c(13, -4, 2, -4, 11, -2, 2, -2, 8), 3, 3))
(D <- eigen(A)$values)
(C <- eigen(A)$vectors)
C %*% diag(D) %*% t(C)
diag(D)

(A1 <- D[1] * C[, 1] %*% t(C[, 1]))
(A2 <- D[2] * C[, 2] %*% t(C[, 2]))
(A3 <- D[3] * C[, 3] %*% t(C[, 3]))
A1 + A2 + A3
all.equal(A, A1+A2+A3)

(A <- matrix(c(13, -4, 2, -4, 11, -2, 2, -2, 8), 3, 3))
(D <- eigen(A)$values)
(C <- eigen(A)$vectors)
A.5 <- C %*% sqrt(diag(D)) %*% t(C)
A.5 %*% A.5
all.equal(solve(A), solve(A.5)%*%solve(A.5)) # extra property

(A <- matrix(c(4, 7, -1, 8, -5, -2, 4, 2, -1, 3, -3, 6), ncol=3))
(A.svd <- svd(A))
U <- A.svd$u
D <- diag(A.svd$d)
V <- A.svd$v
U %*% D %*% t(V)

begin of video 10

[Martyr]

Is there a way to get a copy of all these notes?