They are all non-zero. I have determined that the matrix is diagonalizable and has an inverse. In one part of the problem, I am asked to find the maximum and minimum number of eigenvectors that the matrix could possibly have?
Since A is diagonalizable does that mean it will have n linearly independent eigenvectors. So, is the max and min number of eigenvectors is 8? If they're asking about linearly independent eigenvectors, then you're right, but if they're just asking about eigenvectors, I would say the min and max is always infinite.
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Active 6 years, 6 months ago. Viewed 37k times. Ayoshna Ayoshna 1, 2 2 gold badges 21 21 silver badges 51 51 bronze badges. If it has an inverse, its rank is 8. So it has 8 eigenvectors I think? It doesn't matter whether matrix is invertible or not. Although if a matrix is invertible then it means it is full rank i. This matrix has only one linearly independent eigen vector. Add a comment. Active Oldest Votes.
For any of this, it doesn't matter whether or not the eigenvalues are non-zero. So all vectors other than the basis vectors would become linearly dependent. The relationship between variance and information here, is that, the larger the variance carried by a line, the larger the dispersion of the data points along it, and the larger the dispersion along a line, the more the information it has.
To put all this simply, just think of principal components as new axes that provide the best angle to see and evaluate the data, so that the differences between the observations are better visible. As there are as many principal components as there are variables in the data, principal components are constructed in such a manner that the first principal component accounts for the largest possible variance in the data set.
The second principal component is calculated in the same way, with the condition that it is uncorrelated with i. This continues until a total of p principal components have been calculated, equal to the original number of variables. What you firstly need to know about them is that they always come in pairs, so that every eigenvector has an eigenvalue. And their number is equal to the number of dimensions of the data. For example, for a 3-dimensional data set, there are 3 variables, therefore there are 3 eigenvectors with 3 corresponding eigenvalues.
Without further ado, it is eigenvectors and eigenvalues who are behind all the magic explained above, because the eigenvectors of the Covariance matrix are actually the directions of the axes where there is the most variance most information and that we call Principal Components. And eigenvalues are simply the coefficients attached to eigenvectors, which give the amount of variance carried in each Principal Component. By ranking your eigenvectors in order of their eigenvalues, highest to lowest, you get the principal components in order of significance.
After having the principal components, to compute the percentage of variance information accounted for by each component, we divide the eigenvalue of each component by the sum of eigenvalues. As we saw in the previous step, computing the eigenvectors and ordering them by their eigenvalues in descending order, allow us to find the principal components in order of significance. In this step, what we do is, to choose whether to keep all these components or discard those of lesser significance of low eigenvalues , and form with the remaining ones a matrix of vectors that we call Feature vector.
So, the feature vector is simply a matrix that has as columns the eigenvectors of the components that we decide to keep. This makes it the first step towards dimensionality reduction, because if we choose to keep only p eigenvectors components out of n , the final data set will have only p dimensions.
Continuing with the example from the previous step, we can either form a feature vector with both of the eigenvectors v 1 and v Or discard the eigenvector v 2, which is the one of lesser significance, and form a feature vector with v 1 only:. Discarding the eigenvector v2 will reduce dimensionality by 1, and will consequently cause a loss of information in the final data set.
Because if you just want to describe your data in terms of new variables principal components that are uncorrelated without seeking to reduce dimensionality, leaving out lesser significant components is not needed.
In the previous steps, apart from standardization, you do not make any changes on the data, you just select the principal components and form the feature vector, but the input data set remains always in terms of the original axes i.
In this step, which is the last one, the aim is to use the feature vector formed using the eigenvectors of the covariance matrix, to reorient the data from the original axes to the ones represented by the principal components hence the name Principal Components Analysis. This can be done by multiplying the transpose of the original data set by the transpose of the feature vector. Zakaria Jaadi is a data scientist and machine learning engineer. Check out more of his content on Data Science topics on Medium.
Zakaria Jaadi. April 1, Updated: November 8, Join the Expert Contributor Network. How do you do a PCA?
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