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Scalar and Vector Triple Products

Scalar and Vector Triple Products

Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the resultant is a vector. These rules of the scalar or vector product between two vectors can be applied to multiple vectors to obtain meaningful combinations. The scalar triple product is the dot product of a vector with the cross product of two vectors.
The scalar triple product is the dot product of a vector with the cross product of two vectors.

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Optical systems for efficient triple-matrix-product processing.

R A Athale¹, J N Lee

¹Naval Research Laboratory, Washington, DC 20375, USA.

|September 1, 2009

Summary

This summary is machine-generated.

Three optical systems efficiently evaluate triple-matrix products using space-integrating and hybrid architectures. These systems offer trade-offs in speed and dynamic range for applications like image compression.

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Area of Science:

Optics and Photonics
Information Processing

Background:

Matrix multiplication is fundamental in linear systems analysis and signal processing.
Efficient optical methods are sought for complex matrix operations.

Purpose of the Study:

To develop and evaluate three distinct optical systems for efficient triple-matrix product evaluation ([U][G][V]).
To explore different architectural and device configurations for optimized performance.

Main Methods:

Development of a purely space-integrating optical system.
Development of two hybrid time-and space-integrating optical systems.
Utilizing various input (1-D/2-D spatial light modulators) and output (detector arrays/single-element detectors) devices.

Main Results:

Demonstrated efficient evaluation of triple-matrix products.
Enabled trade-offs between system speed and dynamic range.
Showcased distinct system formats for diverse applications.

Conclusions:

The developed optical systems provide flexible and efficient solutions for triple-matrix product evaluation.
These systems have potential applications in image data compression and orthogonal transforms.