- #26

Mark44

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This is from the basic definition of an eigenvector.excellent ,but please explain

"when a matrix A multiplies an eigenvector, the result is a vector in the same (or possibly opposite) direction"

An eigenvector ##\vec{x}## for a matrix is a nonzero vector for which ##A\vec{x} = \lambda\vec{x}## for some scalar ##\lambda##. From this definition, it can be seen that ##A\vec{x}## results in a vector that is a multiple of ##\vec{x}##, hence it has the same direction or the opposite direction. Any time you multiply a vector by a scalar, the new vector is in the same direction or the opposite direction.