Question

Use partitioned matrices to prove by induction that the product of two lower triangular matrices is also lower triangular. [Hint: \(A\left( {k + 1} \right) \times \left( {k + 1} \right)\) matrix \({A_1}\) can be written in the form below, where \[a\] is a scalar, v is in \({\mathbb{R}^k}\), and Ais a \(k \times k\) lower triangular matrix. See the study guide for help with induction.]

\({A_1} = \left[ {\begin{array}{*{20}{c}}a&{{0^T}}\\0&A\end{array}} \right]\).

Short answer

It is proved that by induction, the product of two lower triangular matrices is also lower triangular.

Step-by-step solution

Show the product of two lower triangular matrices

It is given that the product of two \(1 \times 1\) lower triangular matrices is lower triangular.

Suppose for \({\mathop{\rm n}\nolimits} = k\), the product of two \(k \times k\) lower triangular matrices is also lower triangular and take \({A_1}\) and \({B_1}\)as \(\left( {k + 1} \right) \times \left( {k + 1} \right)\) matrices. The partition matrices are in the form

\({A_1} = \left[ {\begin{array}{*{20}{c}}a&{{0^T}}\\0&A\end{array}} \right],{B_1} = \left[ {\begin{array}{*{20}{c}}b&{{0^T}}\\0&A\end{array}} \right]\).

Here v and w are in \({\mathbb{R}^k}\); \(A\) and B are \(k \times k\) lower triangular matrices, and \(a\), \(b\) are scalars.

Show by induction that the product of two lower triangular matrices is lower triangular

\(A\)and \(B\) must be lower triangular matrices since \({A_1}\) and \({B_1}\) are lower triangular.

\(\begin{array}{c}{A_1}{B_1} = \left[ {\begin{array}{*{20}{c}}a&{{0^T}}\\{\mathop{\rm v}\nolimits} &B\end{array}} \right]\left[ {\begin{array}{*{20}{c}}b&{{0^T}}\\{\mathop{\rm v}\nolimits} &B\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ab + {0^T}{\mathop{\rm w}\nolimits} }&{a{0^T} + {0^T}B}\\{{\mathop{\rm v}\nolimits} b + A{\mathop{\rm w}\nolimits} }&{{\mathop{\rm v}\nolimits} {0^T} + AB}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ab}&{{0^T}}\\{b{\mathop{\rm v}\nolimits} + A{\mathop{\rm w}\nolimits} }&{AB}\end{array}} \right]\end{array}\)

ABis a lower triangular matrix because A and B are \(k \times k\) lower triangular matrices. The form \({A_1}{B_1}\) also indicates lower triangular. Therefore, the statement that lower triangular matrices hold for \(n = k + 1\) is true when \(n = k\).

According to the principle of induction, the statement is true for all \(n \ge 1\).

Thus, it is proved that by induction, the product of two lower triangular matrices is also lower triangular.