Ex. 3.8

[szcf-weiya]

[szcf-weiya]

[henrygatech]

why R22 is a 对角阵?

[szcf-weiya]

why R22 is a 对角阵?

Note that R22 is an upper triangular matrix. Generally, it can be shown that an upper triangular orthogonal matrix $M$ is diagonal.

Write $M$ as column vectors, $[v_1,\ldots,v_p]$, then
$$
\langle v_1, v_i\rangle = 0, \quad i=2,\ldots, p,,
$$
which implies the first elements of $v_i, i=2,\ldots, p$ equal 0. We can continue to show that the second elements of $v_i, i=3,\ldots, p$ are also 0 by noting that
$$
\langle v_2, v_i\rangle = 0, i=3,\ldots,p,.
$$
And so on for other elements. Finally, $M$ would be diagonal.

[ccwwhh101]

这题应该有点问题，应该问必要条件是什么，SVD分解应该不唯一