Inner Product

linear-algebradspsignal-processingmathematics

Abstraction: Inner product concept connecting linear algebra and digital signal processing

Key points:

The inner product of two vectors is the sum of products of corresponding components; result measures correlation — zero means orthogonality.
Complex multiplication is represented by a 2x2 orthogonal matrix whose rows have inner product zero, connecting complex conjugates and perpendicularity.
For vectors with an even number of components, an orthogonal companion can always be constructed by time-reversing and alternating signs (multiply by (-1)^n).
The vector norm (L2) is the square root of a vector's inner product with itself — "Pythagoras extended" to arbitrary dimensions.
Unit vectors (norm = 1) are obtained by dividing each component by the vector's norm; combining N unit vectors into one requires dividing by sqrt(N).
These concepts underlie half-band filter design and perfect-reconstruction filter banks in DSP.

Connections: Inner Product · Linear Algebra · Orthogonality · Digital Signal Processing