Problem of the Week #28 - October 8th, 2012

Status
Not open for further replies.

Chris L T521

Well-known member
Staff member
Thanks to those who participated in last week's POTW!! Here's this week's problem!

-----

Problem: Let $A$ and $B$ be $n\times n$ matrices with real entries. Show that $\langle A\mathbf{u},B\mathbf{v}\rangle = \langle \mathbf{u}, A^TB\mathbf{v}\rangle$ for any vectors $\mathbf{v},\mathbf{w}\in\mathbb{R}^n$, where $\langle\cdot,\cdot\rangle$ denotes the standard inner product on $\mathbb{R}^n$.

-----

Chris L T521

Well-known member
Staff member
This week's question was correctly answered by dwsmith, girdav, and Sudharaka. You can find Sudharaka's solution below.

Take any two vectors $$\mathbf{x},\mathbf{y}\in\mathbb{R}^n$$ where $$\mathbf{x}=(x_1,x_2,\cdots,x_n)^{T}$$ and $$\mathbf{y}=(y_1,y_2,\cdots,y_n)^{T}$$. Then the standard inner product of $$\mathbb{R}^n$$ is defined by,

$\langle\mathbf{x},\mathbf{y}\rangle:=\sum_{i=1}^{n}x_{i}y_{i}=x^{T}y$

Hence we have,

$\langle A\mathbf{u},B\mathbf{v}\rangle =(A\mathbf{u})^{T}(B\mathbf{v})=(\mathbf{u}^{T}A^{T})(B\mathbf{v})~~~~~~~~(1)$

Also,

$\langle \mathbf{u}, A^TB\mathbf{v}\rangle=\mathbf{u}^{T}(A^{T}B\mathbf{v})=(\mathbf{u}^{T}A^{T})(B\mathbf{v})~~~~~~~(2)$

By (1) and (2) we have,

$\langle A\mathbf{u},B\mathbf{v}\rangle = \langle \mathbf{u}, A^TB\mathbf{v}\rangle$

Status
Not open for further replies.