The rank of any matrix A, square or rectangular, can be found from its eigenstructure or that of its product moment matrices. Furthermore, eigenvectors associated with distinct eigenvalues are already orthogonal to begin with. If the eigenvalues are not all distinct, an orthogonal basisâalbeit not a unique oneâcan still be constructed. 13 Moreover, all eigenvalues and eigenvectors are necessarily real. If A (or Aâ²A or AAâ²) is singular with a subset of l k nondistinct eigenvalues, we can still find a mutually orthonormal set of eigenvectors of rank l k by some process, such as the GramâSchmidt orthonormalization process, for the tied block k. If so, their multiplicities are still counted up in finding the rank of A. ![]() However, even if Aâ²A (or AAâ²) is nonsingular, some of the (positive) λ i may be equal to each other. Next, suppose that the symmetric matrix being examined is still of the form Aâ²A or AAâ², where we have adopted this form because A is either rectangular or nonsymmetric. (The number k is the number of positive eigenvalues in Aâ²A or AAâ².) If r( A) < n ⤠m, then the set of either row or column vectors are linearly dependent and r( A) = k is the largest number of linearly independent vectors in A. If r( A) = m< n, then the column vectors are linearly dependent. If r( A) =n and n ![]() ![]() The product-moment matrix will be square and symmetric. First, if A is originally nonsymmetric or rectangular, we can always find the minor product moment ( Aâ²A) or the major product moment ( AAâ²) of A, whichever is of smaller order. For the moment, however, let us set down the procedure for rank determination in a step-by-step way.
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