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Products of the type XTX, XXT, XTDX, ATA and AAT

Let

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Product XTX.

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Product XXT

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Product XTDX where X is a vector and D is a diagonal matrix.

The product

is equal to the following:

XTDX = d1x12 + d2x22 + ... + dnxn2

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Product ATA.

The product ATA 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 ... Cn] where C1, C2, ... ,Cn represent the columns of A. Then

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Product AAT

The product AAT 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, ... ,Rm represent the rows of A. Then