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Steven G. Johnson 2023-02-13 14:37:33 -05:00
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@ -96,6 +96,6 @@ The *nice* case of diagonalization is when you have **orthonormal eigenvectors**
* Unitary matrices Aᴴ=A⁻¹. In this case, we can easily show that |λ|=1, and then prove orthogonality of eigenvectors from the fact that unitary matrices preserve inner products.
**Further reading** [OCW lecture 4](https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/resources/lecture-4-eigenvalues-and-eigenvectors/) and textbook section I.6.
* Complex numbers and vectors: See [18.06 lecture 26](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-26-complex-matrices-fast-fourier-transform/) on complex matrices. This [brief review of complex numbers](https://web.stanford.edu/~boyd/ee102/complex-primer.pdf) (from Stephen Boyd at Stanford) is at about the level of my lecture. There are many more basic reviews, e.g. from [Khan academy](https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex), that you can find online.
* Complex numbers and vectors: See [18.06 lecture 26](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-26-complex-matrices-fast-fourier-transform/) on complex matrices. This [brief review of complex numbers](https://web.stanford.edu/~boyd/ee102/complex-primer.pdf) (from Stephen Boyd at Stanford) is at about the level we need. There are many more basic reviews, e.g. from [Khan academy](https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex), that you can find online.
* Similar matrices are discussed in [18.06 lecture 28](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-28-similar-matrices-and-jordan-form/), and diagonalization is in [18.06 lecture 22](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-22-diagonalization-and-powers-of-a/).
* Hermitian (real-symmetric) matrices: [18.06 lecture 25](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-25-symmetric-matrices-and-positive-definiteness/).