Many proofs in linear algebra are greatly simplified if one can easily deal with the determinants of block matrices, that is, matrices that are subdivided into blocks that are themselves matrices.
A block matrix (also called partitioned matrix) is a matrix of the kindwhere , , and are matrices, called blocks, such that:
and have the same number of rows;
and have the same number of rows;
and have the same number of columns;
and have the same number of columns.
Ideally, a block matrix is obtained by cutting a matrix two times: one vertically and one horizontally. Each of the four resulting pieces is a block.
Example The matrixcan be written as a block matrixwhere
Example The matrixcan be written as a block matrixwhere
An important fact about block matrices is that their multiplication can be carried out as if their blocks were scalars, by using the standard rule for matrix multiplication:
The only caveat is that all the blocks involved in a multiplication (e.g., , , ) must be conformable. For example, the number of columns of and the number of rows of must coincide.
A first result concerns block matrices of the formorwhere denotes an identity matrix, is a matrix whose entries are all zero and is a square matrix.
Block matrices whose off-diagonal blocks are all equal to zero are called block-diagonal because their structure is similar to that of diagonal matrices.
Not only the two matrices above are block-diagonal, but one of their diagonal blocks is an identity matrix. We will call them block-diagonal matrices with identity blocks.
The following proposition holds.
Proposition Let be one of the two block-diagonal matrices with identity blocks defined above. Then,
We first establish the result for the case in which and is , that is, . Suppose is . Then is . We use the definition of determinantwhere is the set of all permutations of the first natural numbers. The term is different from 0 and, in particular, equal to 1 only when . Furthermore, the sign of the permutations in which is determined only by because does not determine any inversion. Thus, we havewhere is the set of all permutations of the first natural numbers. The result for the case in which is not is proved recursively. For example, if is , we haveand analogously for larger dimensions. The proof for the second case, in whichis similar to the one just provided.
A block-upper-triangular matrix is a matrix of the formwhere and are square matrices.
Proposition Let be a block-upper-triangular matrix, as defined above. Then,
Suppose that is and is , so that is and is . In what follows, we will denote by a identity matrix and by an zero matrix. Note thatThus,where: in step we have used the fact that the determinant of a product of square matrices is equal to the product of their determinants; in step we have used the result on the determinant of block-diagonal matrices with identity blocks previously proved; in step we have used the fact that the determinant of an identity matrix is equal to 1.
A block-lower-triangular matrix is a matrix of the formwhere and are square matrices.
Proposition Let be a block-lower-triangular matrix, as defined above. Then,
Suppose that is and is , so that is and is . In what follows, we will denote by a identity matrix and by an zero matrix. Note thatThus, similarly to the previous proof,
Below you can find some exercises with explained solutions.
Use the rules on the determinants of block matrices to compute the determinant of the matrix
The matrix is block-lower triangular:whereTherefore
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