1. A n by n matrix A is called positive semidefinite if x Ax > 0 for any vector x whose length is n. Using this definition, prove that the sum of two positive semidefinite matrices is itself positive semidefinite. Zi 2. Let Z₁ = VTx; ЄR" be the vectors of principal components resulting from PCA (embedding dimension r) for all i = 1,...,n, where V is the matrix of leading eigenvectors of the data covariance matrix of a centered data set D = {(x;)}} 1. Show that the j-th eigenvalue λ; of the data covariance matrix satisfies for all j = 1,..., r. n n (zi) } = λj i=1

1. A n by n matrix A is called positive semidefinite if x Ax > 0 for any vector x whose length is n. Using this definition, prove that the sum of two positive semidefinite matrices is itself positive semidefinite. Zi 2. Let Z₁ = VTx; ЄR" be the vectors of principal components resulting from PCA (embedding dimension r) for all i = 1,...,n, where V is the matrix of leading eigenvectors of the data covariance matrix of a centered data set D = {(x;)}} 1. Show that the j-th eigenvalue λ; of the data covariance matrix satisfies for all j = 1,..., r. n n (zi) } = λj i=1