Deform. Grad. quiz

Question 1

Consider the positive definite symmetric matrix: $M = (\begin{matrix} 16 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{matrix})$

If $U$ is the unique positive definite square root of $M$ , then the components of $U$ are:

U_{11} =

U_{12} =

U_{13} =

U_{21} =

U_{22} =

U_{23} =

U_{31} =

U_{32} =

U_{33} =

Can you guess how many square roots does $M$ have?

M

has:

By statement 2 of the notes, $M$ has a unique positive definite square root. This positive definite square root is a symmetric matrix that has the same eigenvectors of $M$ but whose eigenvalues are the positive square roots of the eigenvalues of M. In that case: $U = (\begin{matrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix})$

$M$ has 8 square roots. The following Mathemtica code can be used to find all possible matrices that are square roots of $M$ along with their eigenvalues and eigenvectors:

M = {{16, 0, 0}, {0, 4, 0}, {0, 0, 1}};
A = {{A11, A12, A13}, {A21, A22, A23}, {A31, A32, A33}};
A.A // MatrixForm
unknowns = Flatten[A]
a = Solve[A.A == M, unknowns]
As = Table[A /. a[[i]], {i, 1, 8}]
Do[Print[As[[i]] // MatrixForm], {i, 1, Length[As]}]
Do[Print[Eigensystem[As[[i]]] // MatrixForm], {i, 1, Length[As]}]

Question 2

The identity matrix has more than one square root that are positive definite symmetric matrices.

Select one
True
False

The statement is false. First note that the identity matrix is symmetric and positive definite. $I = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})$

Using the statements in class, there is only one unique positive definite symmetric matrix that is the square root of $I$ and it happens to be $I$ itself: $I^{2} = I$

$I$ , however, has many other square roots that are also symmetric matrices but are not positive definite. For example:

$A = (\begin{matrix} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})$ is a square root of $I$