Suppose A and B are matrices in lower triangular form show t

Suppose A and B are matrices in lower triangular form, show that A×B is also in lower triangular form. Furthermore, every eigenvalue of A×B has the form of , where is an eigenvalue of A and is an eigenvalue of B (and A×B is the tensor product of A and B).

Solution

Let A and B be the two input upper triangular matrices and C be the product matrix. We have to prove that C is an upper triangular matrix.

Cij = Aik*Bjk

ij element of C is calculated by inner product of i th row of A and jth coulmn of B.

For given i,ji,j this sum will only be nonzero if there are k with i k j (or at least one of Aik,Bkj will vanish) which requires i j Therefore ,C is upper triangular.

Now to prove,  every eigenvalue of A×B has the form of , where is an eigenvalue of A and is an eigenvalue of B

Let Bx = x ;

multiply by matrix A on both sides : ABx = Ax

= Ax

we know that  Ax = x

So, ABx = (x)

ABx = ()x

which means that eigen value of product of AxB is