The lack of uniqueness of B was illustrated in the context of axis rotation in factor analysis. To find the rank of A, we simply count up the number of positive eigenvalues k and observe that r( A) = k ≤ min( m, n) if A is rectangular or r( A) = k ≤ n if A is square.įinally, if A is of product-moment form to begin with, or if A is symmetric with nonnegative eigenvalues, then it can be written–although not uniquely so–as A = B′B. In this case, all eigenvalues are real and nonnegative. If A is nonsymmetric or rectangular, we can find its minor (or major) product moment and then compute the eigenstructure. If A n×n is symmetric, we merely count up the number of nonzero eigenvalues k and note that r( A) = k ≤ n. 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 basis–albeit not a unique one–can still be constructed. 13 Moreover, all eigenvalues and eigenvectors are necessarily real. If A (or A′A or AA′) 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 Gram–Schmidt orthonormalization process, for the tied block k. If so, their multiplicities are still counted up in finding the rank of A. However, even if A′A (or AA′) 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 A′A or AA′, where we have adopted this form because A is either rectangular or nonsymmetric. (The number k is the number of positive eigenvalues in A′A or AA′.) 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
![rank of a matrix rank of a matrix](https://i.ytimg.com/vi/P6ZBG2yFjQc/maxresdefault.jpg)
First, if A is originally nonsymmetric or rectangular, we can always find the minor product moment ( A′A) or the major product moment ( AA′) 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.