Matrix Operations
The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write diag(a1,...,an) for a diagonal matrix whose diagonal entries starting in the upper left corner are a1,...,an. Then, for addition, we have
- diag(a1,...,an) + diag(b1,...,bn) = diag(a1+b1,...,an+bn)
and for matrix multiplication,
- diag(a1,...,an) ยท diag(b1,...,bn) = diag(a1b1,...,anbn).
The diagonal matrix diag(a1,...,an) is invertible if and only if the entries a1,...,an are all non-zero. In this case, we have
- diag(a1,...,an)-1 = diag(a1-1,...,an-1).
In particular, the diagonal matrices form a subring of the ring of all n-by-n matrices.
Multiplying an n-by-n matrix A from the left with diag(a1,...,an) amounts to multiplying the i-th row of A by ai for all i; multiplying the matrix A from the right with diag(a1,...,an) amounts to multiplying the i-th column of A by ai for all i.
Read more about this topic: Diagonal Matrix
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