Complex vectors, complex matrices, inner products, and discrete Fourier Transform
In this lecture, the topic is complex numbers in linear algebra, specifically complex vectors and matrices, inner products, and the discrete Fourier Transform. The length and inner product of complex vectors are defined with the use of Hermitian conjugates. Hermitian matrices are introduced as the complex version of symmetric matrices. The lecture focuses on the Fourier matrix, a complex matrix with orthogonal columns that is crucial in the Fourier transform, and the fast Fourier transform (FFT) that reduces the computation time of the Fourier transform from N squared to N log N. The lecture then concludes with a discussion of the powers of a number w, which is crucial to constructing the Fourier matrix.
Complex vectors, complex matrices, inner products, and discrete Fourier Transform -- Lecture 26. A lecture on Complex numbers in Linear Algebra.
Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 23, 2008). License: Creative Commons BY-NC-SA.
More info at: http://ocw.mit.edu/terms
- Inner Product Java Applet - Java applet to play around to investigate inner products.
- What is the FFT?
- What is the Fast Fourier Transform?
- How do you use the Fast Fourier Transform?
- How do you use complex vectors?
- How do you use complex matrices?
- What are inner products?
- What is the discrete Fourier transform?
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Staff Review
This video is a great lesson explaining Fourier matrices and the Fast Fourier Transform (FFT). Finally, a lesson with complex vectors, complex matrices, and complex numbers in Linear Algebra. Inner Products are discussed as well. A lesson packed full of important ideas.
