Partition of a Matrix

Question

In Linear Algebra, we have been taught that the partition of a matrix $A$ consists of matrices,or blocks. In other words, its elements are matrices. This same, partitioned matrix, however is said to be equal to the original matrix. But their elements are different, as one contains scalars and the other matrices. Please help me understand.

Answer 1

You have to concatenate the matrices. Here's an example: $$A= \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, B = \begin{pmatrix} 10 \\ 20 \end{pmatrix}, C = (100,200), D = 1000,$$ then $$\begin{pmatrix}A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} & \begin{pmatrix} 10 \\ 20 \end{pmatrix} \\ \begin{pmatrix} 100 & 200\end{pmatrix} & 1000 \end{pmatrix} \equiv \begin{pmatrix} 1 & 2 & 10 \\ 3 & 4 & 20 \\ 100 & 2000 & 1000 \end{pmatrix}.$$

Answer 2

In linear algebra, when we talk about the partition or block structure of a matrix A, we're essentially organizing its elements into submatrices or blocks. The key point is that while the individual elements within these blocks might be matrices instead of scalars, the equality between the original matrix A and its partitioned form still holds.

To clarify, the equality is established based on the correspondence of corresponding blocks in the partitioned matrix and the original matrix. Each block in the partitioned matrix is, in fact, a matrix, and the equality is a way of expressing that the entire structure of the partitioned matrix, block by block, is equivalent to the original matrix.

In summary, even though the elements within the blocks are matrices instead of scalars, the equality is upheld by ensuring that the organization and structure of the blocks align with the original matrix A. This concept is often used in various applications, such as solving linear systems or representing certain mathematical operations more conveniently.