Find the polar decompositions A=Q B, as defined in Exercise...

Find the polar decompositions $A=Q B$, as defined in Exercise 8.5.28, of the following matrices: (a) $\left(\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right)$, (b) $\left(\begin{array}{rr}2 & -3 \\ 1 & 6\end{array}\right)$, (c) $\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)$, (d) $\left(\begin{array}{rrr}0 & -3 & 8 \\ 1 & 0 & 0 \\ 0 & 4 & 6\end{array}\right)$, (e) $\left(\begin{array}{rrr}1 & 0 & 1 \\ 1 & -2 & 0 \\ 1 & 1 & 0\end{array}\right)$.

Find the polar decompositions $A=Q B$, as defined in Exercise 8.5.28, of the following matrices:
(a) $\left(\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right)$,
(b) $\left(\begin{array}{rr}2 & -3 \\ 1 & 6\end{array}\right)$,
(c) $\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}0 & -3 & 8 \\ 1 & 0 & 0 \\ 0 & 4 & 6\end{array}\right)$,
(e) $\left(\begin{array}{rrr}1 & 0 & 1 \\ 1 & -2 & 0 \\ 1 & 1 & 0\end{array}\right)$.

Key Concepts

Polar Decomposition

Polar decomposition is a fundamental factorization in matrix analysis where any given matrix is expressed as the product of a unitary (or orthogonal, in the case of real matrices) matrix and a positive semidefinite Hermitian (or symmetric) matrix. This decomposition separates the matrix into a component that preserves lengths (rotations or reflections) and a component that scales vectors, encapsulating both rotational and stretching effects in linear transformations.

Unitary/Orthogonal Matrices

Unitary matrices (or orthogonal matrices when dealing with real entries) are characterized by the property that their conjugate transpose (or transpose in the real case) is also their inverse, ensuring that they preserve the inner product. This preservation reflects in their role in representing rotations and reflections in vector spaces, making them essential in decompositions that require norm conservation.

Positive Semidefinite Matrices

A positive semidefinite matrix is one where all its eigenvalues are nonnegative. In the context of polar decomposition, the positive semidefinite matrix represents the scaling component of the original transformation. Its properties ensure that the associated square root operation is well-defined, providing a symmetric (or Hermitian) factor that captures the inherent 'stretching' effects of the original matrix.

Matrix Square Root

For a given matrix A, the right factor in the polar decomposition is typically obtained by taking the unique positive semidefinite square root of A^T A (or A* A in the complex case). This operation is central to ensuring that the decomposed factors have the desired properties, specifically producing a positive semidefinite matrix that squares back to A^T A, and facilitating the construction of the unitary or orthogonal factor via the relation Q = A multiplied by the inverse of this square root.

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