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let $A \in {M_{n \times m}}$ why every real symmetric matrix has at least one real eigenvalue? .

SD1
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    Do you mean $n \times n$? What does it mean for $A$ to be symmetric when $m \neq n$? BTW, it's a special case of the spectral theorem for normal operators. – shalin Mar 27 '15 at 05:15

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your question can be answered in two parts. one symmetric matrices have real eigenvalues, and two that every matrix has at least one eigenvalue if the base field is algebraically closed. you put the two together to get your answer.

abel
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