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A geometric interpretation of the covariance matrix
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However, before diving into the technical details, it is important to dissertation for dummies review an intuitive understanding of how eigenvectors and eigenvalues uniquely define the covariance matrix, and therefore the dissertation for dummies review of our data. As we saw in figure 3, the covariance matrix defines both the spread varianceand the orientation covariance of our data.
So, if we would like to represent the covariance matrix with a dissertation for dummies review and its magnitude, we should simply try to find the vector that points into the direction of the largest spread of the data, and whose magnitude equals the spread variance in this direction.
If we define this vector asthen the projection of our data onto this vector is obtained asand the variance of the projected data is. Since we are looking for the vector that points into the direction of the largest variance, we should choose its components such that the covariance matrix of the projected data is as large as possible.
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- The covariance matrix defines the shape of the data.
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