To fully demonstrate linear algebra, we have to include the plural. Even a real matrix, eigenvalues and eigenvectors are also often complex.
1. imaginary Review
The real and imaginary imaginary part, the real and imaginary real parts are added when adding the imaginary part and the imaginary part are added, then multiplied by the imaginary number when using \ (I ^ 2 = -1 \) .
In the imaginary plane, imaginary number \ (3 + 2i \) is at coordinates \ ((3, 2) \) a point. Complex \ (z = a + bi \ ) conjugated to \ (\ bar = Z * Z ^ = A-BI \) .
In polar coordinates, it may be written in the form of a plurality of die length and polar angle.
Two die length is multiplied by complex multiplication, addition polar angle.
\[(re^{i\theta})^n=r^ne^{in\theta}\]
2. Hermitian (the Hermitian) matrix and a unitary (Unitary) matrix
This key may be part of one sentence describes: When you transpose of a complex vector or matrix, while their conjugation.
Why do it? One reason is that the special nature of complex vector length. For real vector, which is the square of the length \ (x_1 ^ 2 + \ cdots x_n ^ 2 + \) , but not the square of a complex vector of length \ (^ 2 + Z_1 \ cdots Z_n ^ 2 + \) . For example \ (z = (1, i ) \) square length is not \ (. 1 + 2 ^ 2 ^ I = 0 \) , but should be \ (z \ bar z = 1 ^ 2 + | i ^ 2 | = 2 \) .
我们定义一个新符号，\(\bar z^T=z^H\)，来表示向量的共轭转置，这个符号也可以应用到矩阵中去。
同时，我们也要对向量的内积定义进行一下扩展，但内积为零仍然表明正交。
这时候，向量的顺序就变得重要了。
\[\boldsymbol v^H\boldsymbol u=\bar v_1u_1+\cdots+\bar v_nu_n=(\boldsymbol u^H\boldsymbol v)^*\]
一个厄米特矩阵满足 \(A^H=A\)，每一个实对称矩阵都是厄米特的，因为实数的共轭还是它本身。
如果 \(A^H=A\)，\(\boldsymbol z\) 是任意向量，那么 \(\boldsymbol z^HA\boldsymbol z\) 是实数。
\[(z^HAz)^H=z^HA^Hz=z^HAz\]
来自对角线上的两项都是实数，而来自非对角线上的两项互为共轭，相加之后也为实数。
厄米特矩阵的每个特征值都是实数。
\[A\boldsymbol z=\lambda \boldsymbol z \to \boldsymbol z^HA\boldsymbol z=\lambda \boldsymbol z^H \boldsymbol z\]
上式左边为实数，\(\boldsymbol z^H\boldsymbol z\) 是长度的平方，是正实数，所以特征值也必须为实数。
厄米特矩阵对应于不同特征值的特征向量是正交的。
\[\tag{1}A\boldsymbol z=\lambda \boldsymbol z \to \boldsymbol y^HA\boldsymbol z=\lambda \boldsymbol y^H\boldsymbol z \]
\[\tag{2}A\boldsymbol y=\beta \boldsymbol y \to \boldsymbol z(A\boldsymbol y)^H=\boldsymbol z(\beta \boldsymbol y)^H \to \boldsymbol y^HA\boldsymbol z=\beta \boldsymbol y^H\boldsymbol z\]
比较 (1) 式和 (2) 式可得，两式左边相等，所以右边应该也相等。又由于两个特征值不一样，所以有 \(y^H\boldsymbol z=0\)，两个特征向量正交。
酉矩阵是一个有着标准正交列的方阵。
任意有着标准正交列的矩阵满足 \(U^HU=I\)，如果它还是一个方阵，那么有 \(U^H=U^{-1}\)。
一个酉矩阵乘以任意向量，向量的长度保持不变。
\[\boldsymbol z^HU^HU\boldsymbol z=\boldsymbol z^H\boldsymbol z\]
而且，酉矩阵的所有特征值的绝对值都为 1。
最后，我们来总结一下实数和虚数向量以及矩阵之间的一些概念迁移。
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