Page 207 -

174 ALGEBRA

5.2.1-10. Block matrices.

Let us split a given matrix A ≡ [a ij ](i = 1, 2, ... , m; j = 1, 2, ... , n)ofsize m × n into
separate rectangular cells with the help of (M –1) horizontal and (N –1) vertical lines. Each
cell is a matrix A αβ ≡ [a ij ](i = i α , i α + 1, ... , i α + m α –1; j = j β , j β + 1, ... , j β + n β –1)of
size m α × n β and is called a block of the matrix A.Here i α = m α–1 + i α–1 , j β = n β–1 + j β–1 .
Then the given matrix A can be regarded as a new matrix whose entries are the blocks:
A ≡ [A αβ ](α = 1, 2, ... , M; β = 1, 2, ... , N). This matrix is called a block matrix.
Example 4. The matrix
a 11 a 12 a 13 a 14 a 15
⎛ ⎞
⎜ a 21 a 22 a 23 a 24 a 25 ⎟
⎜ ⎟
A ≡ ⎜ a 31 a 32 a 33 a 34 a 35 ⎟
⎝ a 41 a 42 a 43 a 44 a 45 ⎠
a 51 a 52 a 53 a 54 a 55
can be regarded as the block matrix
A 11 A 12
A ≡
A 21 A 22
of size 2×2 with the entries being the blocks

A 11 ≡ a 11 a 12 a 13 , A 12 ≡ a 14 a 15 ,
a 21 a 22 a 23 a 24 a 25
a 31 a 32 a 33 a 34 a 35
( ) ( )
A 21 ≡ a 41 a 42 a 43 , A 22 ≡ a 44 a 45
a 51 a 52 a 53 a 54 a 55
of size 2×3, 2×2, 3× 3, 3× 2, respectively.
Basic operations with block matrices are practically the same as those with common
matrices, the role of the entries being played by blocks:
1. For matrices A ≡ [a ij ] ≡ [A αβ ]and B ≡ [b ij ] ≡ [B αβ ] of the same size and the same
block structure, their sum C ≡ [C αβ ]=[A αβ + B αβ ] is a matrix of the same size and
the same block structure.
2. For a matrix A ≡ [a ij ]of size m×n regarded as a block matrix A ≡ [A αβ ]of size M ×N,
the multiplication by a scalar is defined by λA =[λA αβ ]= [λa ij ].
3. Let A ≡ [a ik ] ≡ [A αγ ]and B ≡ [b kj ] ≡ [B γβ ] be two block matrices such that the
number of columns of each block A αγ is equal to the number of the rows of the
block B γβ . Then the product of the matrices A and B can be regarded as the block
matrix C ≡ [C αβ ]= [ γ A αγ B γβ ].
4. For a matrix A ≡ [a ij ]of size m×n regarded as a block matrix A ≡ [A αβ ]of size M ×N,
T
the transpose has the form A =[A T ].
βα
5. For a matrix A ≡ [a ij ]of size m×n regarded as a block matrix A ≡ [A αβ ]of size M ×N,
∗
the adjoint matrix has the form A =[A ∗ βα ].
Let A be a nondegenerate matrix of size n × n represented as the block matrix

A ≡ A 11 A 12 ,
A 21 A 22
where A 11 and A 22 are square matrices of size p × p and q × q, respectively (p + q = n).
Then the following relations, called the Frobenius formulas, hold:
–1 –1 –1 –1
11
11
–1 A + A A 12 NA 21 A 11 –A A 12 N
11
A = –1 ,
–NA 21 A 11 N
K –KA 12 A –1
A –1 = –1 –1 –1 22 –1 .
–A A 21 K A + A A 21 KA 12 A 22
22
22
22
–1
–1
–1
–1
Here, N =(A 22 – A 21 A A 12 ) , K =(A 11 – A 12 A A 21 ) ;in the first formula, the matrix
11
22
A 11 is assumed nondegenerate, and in the second formula, A 22 is assumed nondegenerate.