Week1-2 Matrix

2020-03-18 本文已影响0人忻恆

（PS. 此处的公式用的是MathJax）

1) Show using the transpose operator that any square matrix A can be written as the sum of a sym-

metric and a skew-symmetric matrix.

Ans : The square matrix ${\rm A} + {\rm A}^{\rm T}$ is symmetric, and the square matrix ${\rm A} - {\rm A}^{\rm T}$ is skew symmetric.

Using these two matrices, we can write, ${\rm A} = \frac{1}{2}\left( {\rm A}-{\rm A}^{\rm T}\right) +\frac{1}{2}\left( {\rm A}+{\rm A}^{\rm T}\right)$

2) Inner product, dot-product

3) scaler ：标量

4) orthogonal ：正交 ${\rm U}^{\rm T}{\rm V} = 0$ ，perpendicular，垂直

5) the norm : $\lVert {\rm U}\rVert=\left({\rm U}^{\rm T}{\rm U}\right) ^ \frac{1}{2}$

normalize: $\lVert {\rm U}\rVert=1$

6) orthogonal + normalize : orthonormal

7) outer product : ${\rm U}{\rm V}^{\rm T}$ , ${\rm U}{\rm V}$ 均为column vector

8) The trace of a square matrix B, denoted as ${\rm Tr}\,{\rm B}$ ：the sum of the diagonal elements of B

9) Inverse Matrix : ${\rm A}{\rm A}^{\rm -1} ={\rm I}={\rm A}^{\rm -1} {\rm A}$

Week1-2 Matrix

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