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Show that every square matrix can be uniquely expressed as the sum of Hermitian and skew Hermitian matrix.
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written 7.9 years ago by
teamques10
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• modified 6.4 years ago |
Let, A be a given square matrix.
We can write A as-
$ A \; = \; \dfrac{1}{2}(A+A^{\theta}) + \dfrac{1}{2}(A-A^{\theta}) \; = \; say, \; P+Q \; where \; P=\dfrac{1}{2}(A+A^{\theta}) \; and \; Q=\dfrac{1}{2}(A-A^{\theta}) \\ \; \\ \; \\ Now, \;\; P^{\theta}=[\dfrac{1}{2}(A+A^{\theta})]^{\theta} \; = \; \dfrac{1}{2}(A^{\theta}+(A^{\theta})^{\theta}) \; =\; \dfrac{1}{2}(A^{\theta}+A) \; …
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