Prove the properties using matrix notation

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Prove the properties discussed in class for the residual maker, M. We have to prove the properties using matrix notation

The matrix M has the following properties:

1. Symmetric, M = M′.

2. Idempotent, M = MM = M2.

When M is applied twice to any vector, it gives the same result as if it were applied only once. It leaves its image unchanged. Intuitively, the 'shadow of the shadow' is just the shadow itself.

3. MX = 0.

If X is regressed on X instead of y, residuals will be MX instead of My. This model will yield a perfect fit and residuals will be zero.

Reference no: EM133251889

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