= Determinant of a block diagonal matrix
{title2=$\det(A_1\oplus\cdots\oplus A_r)=\prod_j\det A_j$}
In the <Leibniz formula for a determinant> of a <block diagonal matrix>, a term is zero unless its <permutation> maps every block's row indices onto that same block's column indices. The surviving permutations are precisely independent permutations within the blocks; their signs and products of entries factor over the blocks. Summing therefore gives the product of the individual <determinants>. Applied to $tI-T$, this also proves that the <characteristic polynomial> of a block diagonal <linear operator> is the product of the characteristic polynomials of its blocks.
Back to article page
