Several theorems on Matrices

• Perron–Frobenius theorem:
• Kronecker Product:
• Gerschgorin circle theorem:
• Schur Complement:
• Sherman–Morrison–Woodbury formula:

Things to know for matrix theory.

Other than the classical theorems or factorizations from a matrix theory class, I will list some things both interesting and good to know.

• Perron–Frobenius theorem:

Let all the entries of matrix $A \in R^{n\times n}$ are positive (i.e. $A$ is positive square real matrix), then the following holds:

1. $A$ has a real positive eigenvalue r that is strictly larger than any other eigenvalues (they can be complex), that is, $r > \vert \lambda \vert$, where $\lambda$ is eigenvalue of A.

2. $r$ is unique, that is, the multiplicity is 1.
3. There exists an eigenvector associated with $r$ such that all the entries of it are positive.
4. An interesting inequality:

namely, min of column sum $< r <$ max of column sum.

• Kronecker Product:

It has been a long time that I really want to learn and prove the properties of the Kronecker product, as it pops up in so many places in image processing and finite element methods. Here I give several key properties of it, suppose we have

The following holds:

1. Matrix Multiplication/Vectorization:

   $$(A_r \otimes A_c)vec(X) = vec(A_cXA_r^T).$$

2. Transpose and Inversion:

   $$(A_r \otimes A_c)' = A_r^T \otimes A_c^T , \hspace{0.2in} (A_r \otimes A_c)^{-1} = A_r^{-1} \otimes A_c^{-1}.$$

3. Singular value decomposition:

   $$(U_r\Sigma_rV_r^T) \otimes (U_c\Sigma_c V_c^T) = (U_r\otimes U_c) (\Sigma_r\otimes \Sigma_c)(V_r\otimes V_c)^T.$$

• Gerschgorin circle theorem:

• Schur Complement:

• Sherman–Morrison–Woodbury formula: