Positive Definite Block Matrices

  • POTW
  • Thread starter Euge
  • Start date
  • Tags
    Matrix
In summary, a positive definite block matrix is a square matrix composed of sub-matrices, where each sub-matrix is also a square matrix. It is considered positive definite if all of its leading principal minors are positive. Positive definite block matrices have several important properties, including being invertible, having all positive eigenvalues, and being symmetric. They are commonly used in multivariate statistics to represent covariance matrices and can also be used in regression analysis and ANOVA. To check if a block matrix is positive definite, one can use the principal minor test. Positive definite block matrices can also be generalized to higher dimensions, where they are often called positive definite block tensors. The same properties and tests used for block matrices also apply to block tensors.
  • #1
Euge
Gold Member
MHB
POTW Director
2,073
244
Suppose ##A## and ##B## are positive definite complex ##n \times n## matrices. Let ##M## be an arbitrary complex ##n \times n## matrix. Show that the block matrix ##\begin{pmatrix} A & M\\ M^* & B\end{pmatrix}## is positive definite if and only if ##M = A^{1/2}CB^{1/2}## for some matrix ##C## of operator norm ##\|C\| < 1##.
 
Last edited:
  • Like
Likes Greg Bernhardt and topsquark
Physics news on Phys.org
  • #2
The matrix ##\begin{pmatrix}A & M\\M^* & B\end{pmatrix}## is positive definite if and only if
$$\begin{pmatrix}A^{-1/2} & 0\\0 & B^{-1/2}\end{pmatrix} \begin{pmatrix}A & M\\M^* & B\end{pmatrix} \begin{pmatrix} A^{-1/2} & 0\\0 & B^{-1/2}\end{pmatrix} = \begin{pmatrix} I & A^{-1/2}M B^{-1/2}\\ B^{-1/2}M^* A^{-1/2} & I\end{pmatrix}$$ is positive definite. Let ##C = A^{-1/2} M B^{-1/2}##, and write ##C## in SVD ##C = U\Sigma V^*## where ##\Sigma = \operatorname{diag}(\sigma_1,\ldots, \sigma_n)## is the diagonal matrix of singular values of ##C##. Since $$\begin{pmatrix}I & C\\C^* & I\end{pmatrix} = \begin{pmatrix}U & 0\\0 & V\end{pmatrix} \begin{pmatrix} I & \Sigma\\ \Sigma & I\end{pmatrix} \begin{pmatrix}U^* & 0 \\0 & V^*\end{pmatrix}$$ then ##\begin{pmatrix} I & C\\C^* & I\end{pmatrix}## is positive definite if and only if ##\begin{pmatrix}I & \Sigma\\\Sigma & I\end{pmatrix}## is positive definite. The latter matrix is unitarily similar to block sum $$\bigoplus_{i = 1}^n \begin{pmatrix}1 & \sigma_i\\\sigma_i & 1\end{pmatrix}$$ It follows that ##\begin{pmatrix}I & C\\C^* & I\end{pmatrix}## is positive definite if and only if ##\sigma_i^2 < 1## for all ##i##, i.e., ##\|C\| < 1##. Finally, observe that the equation for ##C## is equivalent to ##M = A^{1/2} C B^{1/2}##.
 
  • Like
Likes topsquark

Similar threads

Replies
1
Views
1K
Replies
1
Views
972
Replies
2
Views
746
Replies
6
Views
2K
Replies
10
Views
1K
Replies
2
Views
755
Back
Top