Submatrices and Partitioned Matrices

Let $A_*$ represent an $r \times s$ submatrix of an $m \times n$ matrix $A$ obtained by striking out the $i_1, \ldots , i_{m-r}$th rows and $j_1, \ldots , j_{n-s}$th columns (of $A$), and let $B_*$ represent the $s \times r$ submatrix of $A'$ obtained by striking out the $j_1, \ldots , j_{n-s}$th rows and $i_1, \ldots , i_{m-r}$th columns (of $A'$). Verify that

Verify

Let

represent an $n \times n$ upper block-triangular matrix whose $ij$th block $A_{ij}$ is of dimensions $n_i \times n_j$ ($j \geq i = 1, \ldots , r$). Show that $A$ is upper triangular if and only if each of its diagonal blocks $A_{11}, A_{22}, \ldots , A_{rr}$ is upper triangular.

Let

represent a partitioned $m \times n$ matrix whose $ij$th block $A_{ij}$ is of dimensions $m_i \times n_j$. Verify that

in other words, verify that $A'$ can be expressed as a partitioned matrix, comprising $c$ rows and $r$ columns of blocks, the $ij$th of which is the transpose $A_{ji}'$ of the $ji$th block $A_{ji}$ of $A$. And, letting

represent a partitioned $p \times q$ matrix whose $ij$th block $B_{ij}$ is of dimensions $p_i \times q_j$, verify also that if $c = u$ and $n_k = p_k$ ($k = 1, \ldots , c$) [in which case all of the products $A_{ik}B_{kj}$ ($i = 1, \ldots , r$; $j = 1, \ldots , v$; $k = 1, \ldots , c$), as well as the product $AB$, exist], then

where $F_{ij} = \sum_{k=1}^{c} A_{ik}B_{kj} = A_{i1}B_{1j} + A_{i2}B_{2j} + \cdots + A_{ic}B_{cj}$.