Abstract
This chapter deals with the matrix eigenvalue problem, another major theme in linear algebra as important as the theory of linear equations. This problem is based on the fundamental theorem of algebra, which states that any polynomial with complex coefficients has a complex root. We will give a short proof of it. Though the proof needs some advanced knowledge of analysis, the reader will be able to grasp it with a little effort.
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Notes
- 1.
Moreover, it can be proved that \(\boldsymbol{A}\) is a unitary matrix if \(\boldsymbol{A}\) preserves the norm, using the polarization identity given in Sect. 6.1.
- 2.
Remember to take the complex conjugates of the components of the first vector for the inner product in \(\mathbb {C}^2\).
- 3.
This is an example of diagonalization with a unitary matrix. We will discuss it more generally in Sect. 7.3.
- 4.
The existence of this minimum is not trivial. This is proved from the result that any real-valued continuous function on a bounded closed subset of \(\mathbb {C}\) attains the minimum.
- 5.
The reader familiar with the complex function theory may immediately reach a contradiction. Let \(g(x) = 1/f\left( x\right) \), then \(\left| g\left( x\right) \right| \leqq 1/\left| f\left( a\right) \right| \) by assumption, so \(g\left( x\right) \) would be a bounded holomorphic function defined on the entire complex plane. It must be a constant function by Liouville’s theorem, and hence \(f\left( x\right) \) must also be constant, a contradiction.
- 6.
If \(n = m\), then \(c''_n = \cdots = c''_{m+1} = 0\) and \(h\left( x\right) = -x^m+1\).
- 7.
The formula for a quadratic equation is well known. There also exist the formulas for cubic and quartic equations, but they are not so frequently used.
- 8.
More precisely, in general, a root cannot be described by a finite number of combinations of arithmetic operations and radicals (inverse operations of exponential power) of the coefficients of a given equation. Its proof needs a profound and beautiful theory known as Galois theory.
- 9.
Multiplicity will be explained later in this section.
- 10.
It actually becomes the maximum because there exists \(\boldsymbol{x}\) attaining the supremum. To find such an \(\boldsymbol{x}\) numerically, we solve the maximum problem of the quadratic polynomial \(\left\| {\boldsymbol{Ax}}\right\| ^2\) subject to the constraint of the quadratic equation \(\left\| {\boldsymbol{x}}\right\| ^2=1\). To show only the existence elegantly, we use the facts that the unit sphere in a finite-dimensional normed space is compact and that any real-valued continuous function on a compact set attains the maximum.
- 11.
This range can be replaced by the unit ball \(\left\| {x}\right\| \leqq 1\).
- 12.
Thus, the convergence of matrices is the same as the convergence of vectors discussed in Sect. 6.6.
- 13.
Completeness is a property of a metric space where every Cauchy sequence in the space converges to a point inside it. Consult textbooks on topology.
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Tsukada, M., Kobayashi, Y., Kaneko, H., Takahasi, SE., Shirayanagi, K., Noguchi, M. (2023). Eigenvalues and Eigenvectors. In: Linear Algebra with Python. Springer Undergraduate Texts in Mathematics and Technology. Springer, Singapore. https://doi.org/10.1007/978-981-99-2951-1_7
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DOI: https://doi.org/10.1007/978-981-99-2951-1_7
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