Symmetric matrices and Positive Definite matrices
In this lesson, we learn about the special properties of symmetric matrices and positive definite matrices. The main focus is on the eigenvalues and eigenvectors of symmetric matrices, which are all real and perpendicular, respectively. The eigenvectors can also be chosen to be unit vectors. This leads to a factorization of the matrix into an orthonormal matrix, diagonal matrix, and its transpose. The spectral theorem, which is a famous theorem of linear algebra, tells us that a symmetric matrix can be factored in this way. We also learn that good matrices are those that have real eigenvalues and perpendicular eigenvectors, which are symmetric if real.
Symmetric matrices and Positive Definite matrices -- Lecture 25. Special properties of these matrices and why they are important.
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
- Symmetry Mini-Lecture - This is a mini-lecture demo lesson on Symmetry and symmetric matrices.
- Positive Definite Mini-Lecture - This is a mini-lecture demo lesson on Positive Definite matrices.
- What is a symmetric matrix?
- What is special about Eigenvalues of a symmetric matrix?
- What is special about Eigenvectors of a symmetric matrix?
- What is a positive definite matrix?
- When are Eigenvalues real?
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Staff Review
This lesson introduces symmetric matrices where A = A transpose. Eigenvalues and Eigenvectors have some special properties in these cases. Also, positive definite matrices are discussed. A good explanation of a very interesting class of matrices.
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