Products of the type X^TX, XX^T, X^TDX, A^TA and AA^T

Let

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Product X^TX.

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Product XX^T

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Product X^TDX where X is a vector and D is a diagonal matrix.

The product

is equal to the following:

              X^TDX = d₁x₁² + d₂x₂² + ... + d_nx_n²

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Product A^TA.

The product A^TA is a matrix in which the element in position ( i , j ) consists of the m-dot product of the i-th column and the j-th column of A.

Let an mxn matrix A = [C₁ C₂ ... C_n] where C₁, C₂, ... ,C_n represent the columns of A. Then

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Product AA^T

The product AA^T is a matrix in which the element in position ( i , j ) consists of the m-dot product of the i-th row and the j-th row of A.

Let an mxn matrix

where R₁, R₂, ... ,R_m represent the rows of A. Then