Abstract
In Definition 1.4.2 we introduced the symmetric matrices as those square matrices that are invariant under the transpose operation. That is, those matrices A in \(\mathcal{M}_{n}(\mathbb{K})\) satisfying A = A T. This class of matrices has very important properties: for instance, they are diagonalizable (Theorem 7.4.6) and if the entries of a symmetric matrix are real, then it has only real eigenvalues (Theorem 7.4.5). Here we will study another class of matrices, whose inverses coincide with their transpose. These matrices are called the orthogonal matrices. In this section we restrict ourselves to the case of matrices with real entries. But all the results can be easily extended to the matrices with complex entries.
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Notes
- 1.
If A is a matrix in \(\mathcal{M}_{n}(\mathbb{C}),\) then the assumption of real eigenvalues is not needed and we need to use a unitary matrix instead of the orthogonal one.
- 2.
\(\mathbb{K}\) here is not necessary \(\mathbb{R}\) or \(\mathbb{C}\).
- 3.
Since A is symmetric, we have 〈AX, Y 〉 = 〈X, A T Y 〉 = 〈X, AY〉.
- 4.
We obtain uniqueness only if we assume that the diagonal entries of L are positive.
- 5.
The quadratic optimization problems appear frequently in applications. For instance, many problems in physics and engineering can be stated as the minimization of some energy functions.
- 6.
There are several proofs of this result, here we adapt the one in [7].
- 7.
All the results here remain true if we replace \(\mathbb{R}\) by \(\mathbb{C}\). See Remark 8.1.1.
- 8.
Sometimes referred to as the Hilbert–Schmidt norm and defined as the usual Euclidean norm of the matrix A when it is regarded as a vector in \(\mathbb{R}^{n^{2} }\).
- 9.
In fact, this holds for any matrix norm.
- 10.
This formula yields a technique for estimating the top eigenvalue of A.
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Said-Houari, B. (2017). Orthogonal Matrices and Quadratic Forms. In: Linear Algebra. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-63793-8_8
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