0

So I want to calculate the inverse and determinant of a block matrix ($n\times n$) with diagonal matrices (of same size) as block. Although I have a general idea as to how to calculate the determinant. I can't find anything relevant to find the inverse of such a matrix.

I plan to calculate the determinant by first rearranging the columns, rows to make this matrix a diagonal block matrix, then will calculate the determinant of each of the blocks and multiply them.

Any help in how to calculate the inverse as well as whether my approach to calculating determinant is correct or not is greatly appreciated.

TIA.

Manu S Pillai
  • 101
  • 1
    One rather special thing about diagonal matrices (of the same size) is that they commute. One can therefore apply formulas that work with the scalar entries of a dense matrix to obtain expressions for the determinant and even the inverse of a block matrix with diagonal blocks (provided all blocks are the same size; your Question is not explicit about their size). – hardmath Sep 22 '21 at 16:19
  • Thank you so much for that. I believe now calculating determinant is clear. And yes, all blocks are of the same size, could you please help me with the inverse expression? Is it the 1/det(A) * adjoint(A)? – Manu S Pillai Sep 22 '21 at 17:17
  • @Somos: The wording of the Question describes a block matrix whose blocks are diagonal matrices, not I take it a "block diagonal matrix", whose nonzero entries would be clustered in blocks along the diagonal. – hardmath Sep 23 '21 at 21:15
  • @hardmath Yes, I see more clearly now. You are correct. – Somos Sep 24 '21 at 00:10
  • I don't understand your problem. You do know how to find the inverse and determinant of a general matrix, so all you need to do is apply that to your special matrix. You don't need to do anything special to perform the operations. – Somos Sep 24 '21 at 00:16
  • Hello @Somos, I know how to calculate the inverse and determinant of a matrix, but I was wondering there exist some computationally inexpensive methods exploiting the special structure of this matrix. – Manu S Pillai Sep 24 '21 at 11:23

1 Answers1

1

The direct sum of $n$ copies of the field of scalars are represented by $n\times n$ diagonal matrices. These matrices form a commutative ring with unit denoted by $R$. The matrices

block matrix ($n\times n$) with diagonal matrices (of same size) as block

you describe are matrices can also be interpreted as matrices with elements in $R$. The usual matrix operations of addition, multiplication and division give the same results if the matrices are regarded as matrices with elements in $R$.

You asked

Any help in how to calculate the inverse as well as whether my approach to calculating determinant is correct or not is greatly appreciated.

A similar result holds for determinants but in the final step you take the determinant of the resulting matrix in $R$ which is the product of all of the diagonal elements.

Somos
  • 37,457
  • 3
  • 35
  • 85