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Are eigenvectors always Orthonormal?
In general, for any matrix, the eigenvectors are NOT always orthogonal. But for a special type of matrix, symmetric matrix, the eigenvalues are always real and the corresponding eigenvectors are always orthogonal. However two eigenvectors corresponding to different eigenvalues are orthogonal.
Are PCA axis orthogonal?
PCA is defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on.
Are eigenvectors of covariance matrix orthogonal?
Because the eigenvectors of the covariance matrix are orthogonal to each other, they can be used to reorient the data from the x and y axes to the axes represented by the principal components.
What is an orthonormal basis of eigenvectors?
An orthonormal set must be linearly independent, and so it is a vector basis for the space it spans. Such a basis is called an orthonormal basis. Another instance when orthonormal bases arise is as a set of eigenvectors for a symmetric matrix.
Why are eigenvectors orthogonal to each other?
If v is an eigenvector for AT and if w is an eigenvector for A, and if the corresponding eigenvalues are different, then v and w must be orthogonal. Of course in the case of a symmetric matrix, AT = A, so this says that eigenvectors for A corresponding to different eigenvalues must be orthogonal.
Why do eigenvectors have to be orthogonal in PCA?
In the latter case, the eigenspace for that repeated eigenvalue can contain eigenvectors which are not orthogonal. So your instinct to question why the eigenvectors have to be orthogonal was a good one; if there are repeated eigenvalues they may not be!
What kind of matrix is PCA applied to?
The PCA is applied on this symmetric matrix, so the eigenvectors are guaranteed to be orthogonal. In the context of PCA: it is usually applied to a positive semi-definite matrix, such as a matrix cross product, X ′ X, or a covariance or correlation matrix.
Do you have an intuitive understanding of eigenvectors?
There are many articles out there explaining PCA and its importance, though I found a handful explaining the intuition behind Eigenvectors in the light of PCA. This article aims to give a visual and intuitive understanding of Eigenvectors, such that it makes you better equipped to understand PCA.
Why are the eigenvectors of a symmetric matrix orthogonal?
This has repeated eigenvalue 1 and any two non-zero vectors are eigenvectors, orthogonal or not. (Thinking out such special cases is often a good way to spot counter-examples.) If a symmetric matrix has a repeated eigenvalue, we can choose to pick out orthogonal eigenvectors from its eigenspace.