(a) Prove that every positive definite matrix K has a unique...

(a) Prove that every positive definite matrix $K$ has a unique positive definite square root, i.e., a matrix $B>0$ satisfying $B^2=K$. (b) Find the positive definite square roots of the following matrices: (i) $\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right)$ (ii) $\left(\begin{array}{rr}3 & -1 \\ -1 & 1\end{array}\right)$ (iii) $\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 9\end{array}\right)$, (iv) $\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right)$.

(a) Prove that every positive definite matrix $K$ has a unique positive definite square root, i.e., a matrix $B>0$ satisfying $B^2=K$.
(b) Find the positive definite square roots of the following matrices:
(i) $\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right)$
(ii) $\left(\begin{array}{rr}3 & -1 \\ -1 & 1\end{array}\right)$
(iii) $\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 9\end{array}\right)$,
(iv) $\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right)$.

Key Concepts

Positive Definite Matrix

A positive definite matrix is a symmetric matrix, meaning it equals its own transpose, and all its eigenvalues are positive. This property is essential in ensuring that certain quadratic forms are always positive, which has important implications in optimization, stability analysis, and many areas of applied mathematics.

Matrix Square Root

The matrix square root of a given matrix is another matrix which, when multiplied by itself, yields the original matrix. In the context of positive definite matrices, the square root is itself positive definite. This generalizes the notion of a square root from scalars to matrices, allowing one to perform operations such as solving matrix equations and analyzing differential equations.

Uniqueness of the Positive Definite Square Root

For every positive definite matrix, there exists a unique positive definite matrix that acts as its square root. The uniqueness comes from the properties of symmetric matrices and the manner in which the square roots of the eigenvalues (which are positive) are taken, ensuring that different potential square roots do not exist within the realm of positive definiteness.

Spectral Theorem

The spectral theorem states that a real symmetric matrix can be diagonalized using an orthogonal matrix. This means the matrix can be expressed as Q?Q^T, where ? is a diagonal matrix of eigenvalues. This theorem is fundamental in proving the existence and uniqueness of the matrix square root since it allows one to take the square root of each eigenvalue in ? and then reassemble the square root matrix as Q?^(1/2)Q^T.

Diagonalization

Diagonalization is the process of converting a matrix into a diagonal form via a similarity transformation. For symmetric matrices, this involves an orthogonal transformation, which simplifies the computation of matrix functions like the square root by reducing the problem to applying the function directly to the diagonal elements (i.e., the eigenvalues) of the matrix.

Transcript

Please give Ace some feedback