SOCR ≫ DSPA ≫ Topics ≫

1 How is Matrix multiplication defined?

Validate that \((A_{k,n}\times B_{n,m})^T = (B^T_{m,n})\times (A^T_{n,k})\), both using math notation (first principles) and using R functions for some example matrices.

2 Scalar vs. Matrix Multiplication

Demonstrate the differences between the scalar multiplication (\(*\)) and matrix multiplication (\(\%*\%\)).

3 Matrix Equations

Write a simple matrix solver (\(b = Ax\), i.e.,\(x= A^{-1}b\)) and validate its accuracy using the R command solve(A,b). Solve this equation: \[ \begin{align*} 2a - b + 2c &= 5\\ -a - 2b + c &= 3\\ a + b - c &= 2 \end{align*}. \]

4 Least Square Estimation

Use the SOCR Knee Pain dataset, extract the RB = Right-Back locations \((x,y)\), and fit in a linear model for vertical location (\(y\)) in terms of the horizontal location (\(x\)). Display the linear model on top of the scatter plot of the paired data.

5 Matrix manipuation

Create a matrix \(A\) with elements seq(1, 15, length = 6) and argument nrow = 3, add a row to this matrix; add two columns to A to obtain a new matrix \(C_{4,4}\). Then generate a diagonal matrix \(D\) with \(dim = 4\) and elements rnorm(4,0,1). Apply element wise addition, subtraction, multiplication and division to the matrices \(C\) and \(D\). Apply matrix multiplication to \(D\) and \(C\). Obtain the inverse of \(C\) and compare the result to ginv().

6 Matrix Transpose

Validate the multiplicative transpose formula (\((A_{k,n}\cdot B_{n,m})^{T} = (B_{n,m})^{T}\cdot (A_{k,n})^{T}\)), both using math notation, as well as using R calls, e.g., you can try A = matrix(1:6,nrow=3), B = matrix(2:7, nrow = 2).

7 Sample Statistics

Use the SOCR Data Iris Sepal Petal Classes to compute sample mean and variance of each variables. Then calculate the sample covariance - use both math notation and R built-in functions.

8 Eigenvalues and Eigenvectors

Generate a random matrix with A = matrix(rnorm(9,0,1),nrow = 3), compute the eigenvalues and eigenvectors for \(A\). Then try to solve this equation \(det(A-\lambda I) = 0\), where \(\lambda\) is a vector of length \(3\). Compare \(\lambda\) and the eigenvalues you computed above.

Example of manual and automated calculations of eigen-spectra (eigen-values and eigen-vectors) 

<code>
# A <- matrix(rnorm(9,0,1),nrow = 3); A  # define a random design matrix, may generate complex solutions
A <- matrix(c(0,1/4,1/4,3/4,0,1/4,1/4,3/4,1/2),3,3,byrow=T); A
eigen_spectrum <- eigen(A); eigen_spectrum  # compute the eigen spectrum (eigen-values, $l$, and eigen-vectors, $v$), $ A \times v = l \times v$.
B <- A-eigen(A)$values*diag(3); B   # compute B = (A - eigen_value \times I)
det(A-eigen(A)\$values*diag(3)) # verrify that the det(A-eigen(A)\$values*diag(3)) is not trivial ($0$)
A%*%eigen(A)$vector - eigen(A)$value*diag(3) # validate that $ A \times v = l \times v$.
all.equal(A,  eigen(A)\$vector %*% diag(eigen(A)\$values) %*% solve(eigen(A)$vector))  # compare A = v*l*inv(v)
all.equal(diag(3),  A%*%eigen(A)$vector - eigen(A)$values * eigen(A)$vector)
# The last line compares I == AV - lambda\*v, mind the $*$ and $%*%$ scalar and matrix operators
</code>

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