A complex \(k\)-dimensional vector is an ordered \(k\)-tuple of complex numbers: \[ \mathbf{a} = (a_1, a_2, \ldots, a_k). \] Sometimes we will write \(\mathbf{a}\) as a column vector: \[ \mathbf{a} = \begin{bmatrix}a_1 \\ a_2 \\ \vdots \\ a_k\end{bmatrix}. \]
The set of all complex \(k\)-dimensional vectors is denoted \(\mathbb{C}^k\).
We can add vectors componentwise: \[ \mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2, \ldots, a_k + b_k), \]
We can also multiply a vector by a complex scalar \(c\): \[ c \mathbf{a} = (c a_1, c a_2, \ldots, c a_k). \]
The zero vector is \(\mathbf{0} = (0, 0, \ldots, 0)\).
All these operations satisfy the usual properties of vector addition and scalar multiplication, making \(\mathbb{C}^k\) a vector space.
The inner product of two vectors is defined by \[ \langle \mathbf{a}, \mathbf{b} \rangle = a_1 \overline{b_1} + a_2 \overline{b_2} + \cdots + a_k \overline{b_k}, \] where \(\overline{b_j}\) denotes the complex conjugate of \(b_j\).
The norm of a vector is defined by \[ \| \mathbf{a} \| = \sqrt{\langle \mathbf{a}, \mathbf{a} \rangle} = \sqrt{|a_1|^2 + |a_2|^2 + \cdots + |a_k|^2}. \]
Properties of inner products and norms
The inner product has the following properties:
Linearity in the first argument: For all \(\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{C}^k\) and \(c, d \in \mathbb{C}\), we have \[ \langle c\mathbf{a} + d\mathbf{b}, \mathbf{c} \rangle = c \langle \mathbf{a}, \mathbf{c} \rangle + d \langle \mathbf{b}, \mathbf{c} \rangle. \]
Conjugate linearity in the second argument: For all \(\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{C}^k\) and \(c, d \in \mathbb{C}\), we have \[ \langle \mathbf{a}, c\mathbf{b} + d\mathbf{c} \rangle = \overline{c} \langle \mathbf{a}, \mathbf{b} \rangle + \overline{d} \langle \mathbf{a}, \mathbf{c} \rangle. \]
Conjugate symmetry: For all \(\mathbf{a}, \mathbf{b} \in \mathbb{C}^k\), we have \[ \langle \mathbf{a}, \mathbf{b} \rangle = \overline{\langle \mathbf{b}, \mathbf{a} \rangle}. \]
The norm has the following properties:
Positive definiteness: For all \(\mathbf{a} \in \mathbb{C}^k\), we have \(\| \mathbf{a} \| \geq 0\), and \(\| \mathbf{a} \| = 0\) if and only if \(\mathbf{a} = \mathbf{0}\).
Homogeneity: For all \(\mathbf{a} \in \mathbb{C}^k\) and \(c \in \mathbb{C}\), we have \(\| c\mathbf{a} \| = |c| \| \mathbf{a} \|\).
Lemma 3.1 (Polarization Identity)
For any \(\mathbf{a}, \mathbf{b} \in \mathbb{C}^k\), we have \[ \| \mathbf{a} + \mathbf{b} \|^2 = \| \mathbf{a} \|^2 + 2\operatorname{Re}{\langle \mathbf{a}, \mathbf{b} \rangle} + \| \mathbf{b} \|^2, \] where \(\operatorname{Re}{z}\) denotes the real part of a complex number \(z\).
Cauchy-Schwarz Inequality
For any \(\mathbf{a}, \mathbf{b} \in \mathbb{C}^k\), we have \[ |\langle \mathbf{a}, \mathbf{b} \rangle| \leq \| \mathbf{a} \| \| \mathbf{b} \|. \]
Triangle Inequality
For any \(\mathbf{a}, \mathbf{b} \in \mathbb{C}^k\), we have \[ \| \mathbf{a} + \mathbf{b} \| \leq \| \mathbf{a} \| + \| \mathbf{b} \|. \]
Pythagorean Theorem
If \(\mathbf{a}_1,\mathbf{a}_2, \ldots, \mathbf{a}_n\) are mutually orthogonal vectors in \(\mathbb{C}^k\), then \[ \| \mathbf{a}_1 + \mathbf{a}_2 + \cdots + \mathbf{a}_n \|^2 = \| \mathbf{a}_1 \|^2 + \| \mathbf{a}_2 \|^2 + \cdots + \| \mathbf{a}_n \|^2. \]
A vector \(\mathbf{u}\) is normalized, or is a unit vector if \(\| \mathbf{u}\|=1\).
A collection of nonzero vectors \(\{\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n\}\) is an orthogonal set if \(\langle \mathbf{a}_j, \mathbf{a}_k \rangle = 0\) for all \(j \neq k\). It is an orthonormal set if it is an orthogonal set and each \(\mathbf{a}_j\) is a unit vector.
Theorem
All orthogonal sets of nonzero vectors are linearly independent.
Theorem 3.1
Let \(\{\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_k\}\) be an orthonormal set of vectors in \(\mathbb{C}^k\). Then for any \(\mathbf{a} \in \mathbb{C}^k\), we have
\[ \mathbf{a} = \langle \mathbf{a}, \mathbf{u}_1 \rangle \mathbf{u}_1 + \langle \mathbf{a}, \mathbf{u}_2 \rangle \mathbf{u}_2 + \cdots + \langle \mathbf{a}, \mathbf{u}_k \rangle \mathbf{u}_k, \]
and \[ \| \mathbf{a} \|^2 = |\langle \mathbf{a}, \mathbf{u}_1 \rangle|^2 + |\langle \mathbf{a}, \mathbf{u}_2 \rangle|^2 + \cdots + |\langle \mathbf{a}, \mathbf{u}_k \rangle|^2. \]