東京大学 情報理工学系研究科 コンピュータ科学専攻 2016年2月実施 問題1

Author

kainoj

Description

A unit lower triangular matrix is a lower triangular matrix whose diagonal elements are all equal to .

Answer the following questions.

(1) Suppose that and are lower triangular matrices. Prove that the product of them, , is also a lower triangular matrix.

(2) Suppose that and are unit lower triangular matrices. Prove that is also a unit lower triangular matrix.

(3) Compute the inverse matrices of

respectively.

(4) Suppose that an invertible matrix is decomposed in two ways as , where and are upper triangular matrices, and and are unit lower triangular matrices. Prove that and .

You can use the following facts:

• (i) The inverse of an upper triangular matrix, if it exists, is also an upper triangular matrix.
• (ii) The inverse of a unit lower triangular matrix always exists, and it is also a unit lower triangular matrix.

Kai

(1)

A lower triangular matrix has for . Let be a lower triangular matrices. Let's look closer at entries above diagonal of , i.e. for :

Every item of the summation yields either or . Thus, for .

The same another way:

(2)

Let be a unit lower triangular matrices, i.e for . We know from question (1), that is lower triangular.

Let's examine diagonal items, :

(3)

Inverse of a unit lower triangular matrix is also unit lower triangular. So, we need to find only one entry of the inverse of , such that:

Obviously, . Simillary, :

Here, , , .

(4)

Prove that if has two different LU decomposition, that is and are lower unit matrices, then and .

Assume that inverses of exist. Start with and multiply right-hand by and left-hand by :

We know from question (2) that left-hand side of the last equation is lower unit triangular matrix. In similar manner, we can show that right-hand is upper triangular. Lower and upper triangular matrices can be equal iff they are both diagonal. Moreover, since has ones on diagonal, so must have. We conclude:

That is, and .