Every real matrix can be decomposed as: where is a orthogonal matrix, is a matrix, having non-zero elements only on the diagonal, and is a orthogonal matrix.

We know from the previous post that a symmetric matrix is digonalisable, and can be diagonalised by a orthogonal matrix. In our case happens to be a symmetric matrix.

Therefore, , and . Real symmetric matrices have real eigenvalues and additionally because:

Because is grater or equal to for some vector , it follows that when (i.e. is in the null space of ) and otherwise.

For , multiplying by and considering that and are orthogonal unit vectors, we get:

=>

=>

Denoting we get: =>

is the rank of the matrix.

The set of vectors is extended by the set of orthogonal vectors to form a basis in .

The set of vectors is extended by the set of orthogonal vectors to form a basis in .

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