Inner Product of Two Vectors

The inner product (also known as the dot product) of two vectors is a scalar quantity obtained by multiplying corresponding components of the vectors and summing the products. For two vectors $\mathbf{a} = (a_1, a_2, \ldots, a_n)$ and $\mathbf{b} = (b_1, b_2, \ldots, b_n)$, the inner product is defined as:

$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \cdots + a_nb_n = \sum_{i=1}^{n} a_i b_i$

Example:

Let $\mathbf{a} = (2, 3, 4)$ and $\mathbf{b} = (1, 0, 5)$.

The inner product is:

$\mathbf{a} \cdot \mathbf{b} = (2 \cdot 1) + (3 \cdot 0) + (4 \cdot 5) = 2 + 0 + 20 = 22$

So, the inner product of $\mathbf{a}$ and $\mathbf{b}$ is 22.

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Inner Product of Two Matrices

The inner product of two matrices is a generalization of the vector inner product. For two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$, both of size $m \times n$, the inner product is the sum of the products of corresponding elements:

$A : B = \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} b_{ij}$

Example:

Let:

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

The inner product of matrices $A$ and $B$ is:

$A : B = (1 \cdot 5) + (2 \cdot 6) + (3 \cdot 7) + (4 \cdot 8) = 5 + 12 + 21 + 32 = 70$

So, the inner product of $A$ and $B$ is 70.

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Summary:

• The inner product of two vectors results in a scalar by multiplying corresponding elements and summing them.
• The inner product of two matrices results in a scalar by summing the products of corresponding elements across both matrices.

Would you like more details on any part of this?

Here are 5 related questions:

1. How is the inner product used in geometry or physics?
2. What are the properties of the inner product?
3. Can we generalize the inner product to complex vectors?
4. How is the inner product related to vector norms?
5. What is the outer product, and how is it different from the inner product?

Tip: The inner product of two orthogonal vectors is always zero!