Math Tools¶
| date: | 2020/5/10 |
|---|
This note shows the collection of math tools we will use very often.
Matrix Exponent¶
The Taylor series centered at zero for the exponential function \(e^{x}\) (also called exponential function of Maclaurin series)
\[e^{x} = \sum_{n=0}^{\infty}
\frac{x^n}{n!}
= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]
Inspired by this Maclaurin series, we define matrix exponent as
\[e^\mathbf{A}
\equiv \displaystyle \sum_{n=0}^{\infty}
\frac{\mathbf{A}^n}{n!}
= 1 + \mathbf{A} + \frac{\mathbf{A}^2}{2!} + \frac{\mathbf{A}^3}{3!} + \cdots\]
Euler’s Identity¶
If \(\mathbf{A}^2=\mathbf{I}\) …
Logarithm of a matrix¶
A complex matrix has a logarithm if and only if it is invertible…