Partition Matrices
A block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.
Partitoned matrices appear in most modern applications of linear algebra because the notation highlights essential structures of matrices. In particular, partitoned matrices play an essential role in the finite element method. Partititoning a matrix is a generalization to used previously a list of columns or rows. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Especially when dimensions of a matrix are large, it may be beneficial to view a matrix as combined from smaller submatrices. If we simultaneously partition adjacent rows and adjacent columns of a matrix into groups, this partitions the matrix into submatrices or blocks, resulting in a representation of the matrix as a partitioned or block matrix.
If matrices A and B are the same size and are partitioned in exactly the same way, then it is natural to make the same partition of the ordinary matrix sum A + B, and sum corresponding blocks. Similarly, one can subtract the partitioned matrices. Multiplication of a partitioned matrix by a scalar is also computed block by block.
It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions" between two matrices A and B such that all submatrix products that will be used are defined in usual row-column rule.
B = {{6}, {-3}, {1}, {4}, {-1}}
M.B

The Schur complement is named after Issai Schur (1875--1941), who introduced it in 1917 (I. Schur, Potenzreihen im Innern des Einheitskreises, J. Reine Angew. Math., 147, 1917, 205–232). The USA mathematician Emilie Virginia Haynsworth (1916--1985) was the first in 1968 paper to call it the Schur complement. The Schur complement is a key tool in the fields of numerical analysis, statistics and matrix analysis.
Issai Schur was a Russian mathematician (he was born in Mogilev, now Belarus) who worked in Germany for most of his life. Issai spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at the University of Bonn, professor in 1919. As a student of Ferdinand Georg Frobenius (1849--1917), he worked on group representations. He is perhaps best known today for his result on the existence of the Schur decomposition, which we will discuss later.
In 1922 Schur was elected to the Prussian Academy, proposed by Planck, the secretary of the Academy. From 1933 events in Germany made Schur's life increasingly difficult. Schur considered himself as a German, not a Jew, but the Nazis had different opinion. Later in 1935 Schur was dismissed from his chair in Berlin but he continued to work there suffering great hardship and difficulties. Schur left Germany for Palestine in 1939, broken in mind and body, having the final humiliation of being forced to find a sponsor to pay the 'Reichs flight tax' to allow him to leave Germany. Without sufficient funds to live in Palestine he was forced to sell his beloved academic books to the Institute for Advanced Study in Princeton.
DD = {{-3, 1, 1}, {4, 2, -3}, {3, -1, 2}}
A = {{1, -2}, {3, 1}}
B = {{3, -1, 4}, {-2, 4, 2}}
CC = {{5, 4}, {2, -3}, {0, 1}}
MA = (DD - CC.Inverse[A].B)*7
Block Matrix Determinant
For a block matrixAll of these equations can be directly verified by matrix multiplication.
A2 = {{1, 2, 3}, {2, 3, 4}}
A3 = {{-1, 1}, {2, 3}, {3, 4}}
A4 = {{2, 1, -1}, {1, 1, 1}, {2, 3, 4}}
M = ArrayFlatten[{{A1, A2}, {A3, A4}}]
Block Matrix Inversion
To find the inverse matrix to a block matrixB = {{2, -1, 3}, {-1, 1, 2}}
CC = {{-3, 2}, {1, -3}, {2, 1}}
DD= {{2, 4, -6}, {-3, -11, 18}, {-2, -8, 13}}
MA = DD - CC.Inverse[A].B
Inverse[DD] + Inverse[DD].CC.Inverse[MD].B.Inverse[DD]
Inverse[A] + Inverse[A].B.Inverse[MA].CC.Inverse[A]
-Inverse[MA].CC.Inverse[A]
-Inverse[A].B.Inverse[MA]
Out[11]= {{82/1089, 134/363, -(53/99)}, {7/363, 38/121, -(17/33)}, {2/33, 3/11, -(1/3)}}
Out[12]= {{46/363, -(2/363)}, {-(74/1089), 19/1089}}
Out[13]= {{395/1089, -(175/1089)}, {47/363, 140/363}, {4/33, 7/33}}
Out[14]= {{-(109/363), -(4/121), -(4/33)}, {128/1089, -(83/363), 38/99}}
Inverse[M]*3501
Block Diagonal Matrices
A block diagonal matrix is a block matrix that is a square matrix, and having main diagonal blocks square matrices, such that the off-diagonal blocks are zero matrices. A block diagonal matrix M has the formFor the determinant and trace, the following properties hold
A4 = {{2, 1, -1}, {1, 1, 1}, {2, 3, 4}}
zero23 = ConstantArray[0, {2, 3}]
zero32 = ConstantArray[0, {3, 2}]
A = ArrayFlatten[{{A1, zero23}, {zero32, A4}}]
Block Tridiagonal Matrices
A block tridiagonal matrix is another special block matrix, which is just like the block diagonal matrix a square matrix, having square matrices (blocks) in the lower diagonal, main diagonal and upper diagonal, with all other blocks being zero matrices. It is essentially a tridiagonal matrix but has submatrices in places of scalars. A block tridiagonal matrix M has the formwhere Ak, Bk and Ck are square sub-matrices of the lower, main and upper diagonal respectively.
Product of two block matrices
Suppose that we are given two matrices Am×p and Bp×n that are partitioned as follows:Direct Sum
Direct Product