Abstract
In this chapter we use the determinant map in order to assign to every square matrix a unique polynomial that is called the characteristic polynomial of the matrix. This polynomial contains important information about the matrix. For example, one can read off the determinant and thus see whether the matrix is invertible. Even more important are the roots of the characteristic polynomial, which are called the eigenvalues of the matrix.
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Notes
- 1.
Arthur Cayley (1821–1895) showed this theorem in 1858 for \(n=2\) and claimed that he had verified it for \(n=3\). He did not feel it necessary to give a proof for general n. Sir William Rowan Hamilton (1805–1865) proved the theorem for the case \(n=4\) in 1853 in the context of his investigations of quaternions. One of the first proofs for general n was given by Ferdinand Georg Frobenius (1849–1917) in 1878. James Joseph Sylvester (1814–1897) coined the name of the theorem in 1884 by calling it the “no-little-marvelous Hamilton-Cayley theorem”.
- 2.
Oskar Perron (1880–1975).
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Liesen, J., Mehrmann, V. (2015). The Characteristic Polynomial and Eigenvalues of Matrices. In: Linear Algebra. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-24346-7_8
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DOI: https://doi.org/10.1007/978-3-319-24346-7_8
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