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If we think of matrices as multidimensional generalizations of numbers, we may draw useful analogies between numbers and matrices. Not least of these is an analogy between positive numbers and positive definite matrices.
Just as we can take square roots of positive numbers, so can we take “square roots” of positive definite matrices.
A symmetric matrix x is said to be:
These definitions may seem abstruse, but they lead to an intuitively appealing result.
A symmetric matrix x is:
It is useful to think of positive definite matrices as analogous to positive numbers and positive semidefinite matrices as analogous to nonnegative numbers. The essential difference between semidefinite matrices and their definite analogues is that the former can be singular whereas the latter cannot. This follows because a matrix is singular if and only if it has a 0 eigenvalue.
Nonnegative numbers have real square roots. Negative numbers do not. An analogous result holds for matrices. Any positive semidefinite matrix h can be factored in the form for some real square matrix k, which we may think of as a matrix square root of h. The matrix k is not unique, so multiple factorizations of a given matrix h are possible. This is analogous to the fact that square roots of positive numbers are not unique either. If h is nonsingular (positive definite), k will be nonsingular. If h is singular, k will be singular.
A particularly easy factorization to perform is one known as the Cholesky factorization. Any positive semidefinite matrix has a factorization of the form where g is a lower triangular matrix. Solving for g is straightforward. Suppose we wish to factor the positive definite matrix
[1] 
A Cholesky factorization takes the form
[2] 
By inspection, Also by inspection, Since we already have , we conclude . Proceeding in this manner, we obtain a matrix g in 6 steps:
[3] 
The above example illustrates a Cholesky algorithm, which generalizes for higherdimensional matrices. Our algorithm entails two types of calculations:
For a positive definite matrix h, all diagonal elements will be nonzero. Solving for each entails taking the square root of a nonnegative number. We may take either the positive or negative root. Standard practice is to take only positive roots. Defined in this manner, the Cholesky matrix of a positive definite matrix is unique.
The same algorithm applies for singular positive semidefinite matrices h, but the result is not generally called a Cholesky matrix. This is just an issue of terminology. When the algorithm is applied to the singular h, at least one diagonal elements equals 0. If only the last diagonal element equals 0, we can obtain g as we did in our example. If some other diagonal element equals 0, offdiagonal element will be indeterminate. We can set such indeterminate values equal to any value within an interval [–a, a], for some a 0.
[4] 
Performing the first four steps of our algorithm above, we obtain
[5] 

In the fifth step, we multiply the second row of g by the third column of g’ to obtain
We already know , so we have
[7]  
[8] 
which provides us with no means of determining . It is indeterminate, so we set it equal to a variable x and proceed with the algorithm. We obtain
[9] 
For the element to be real, we can set x equal to any value in the interval [–3,3]. The interval of acceptable values for indeterminate components will vary, but it will always include 0. For this reason, it is standard practice to set all indeterminate values equal to 0. With this selection, we obtain
[10] 
We can leave g in this form, or we can delete the second column, which contains only 0’s. The resulting 32 matrix provides a valid factorization of h since
[11] 
If a matrix h is not positive semidefinite, our Cholesky algorithm will, at some point, attempt to take a square root of a negative number and fail. Accordingly, the Cholesky algorithm is a means of testing if a matrix is positive semidefinite.
In exact arithmetic, the Cholesky algorithm will run to completion with all diagonal elements > 0 if and only if the matrix h is positive definite. It will run to completion with all diagonal elements 0 and at least one diagonal element = 0 if and only if the matrix h is singular positive semidefinite.
Things are more complicated if arithmetic is performed with rounding, as is done on a computer. Offdiagonal elements are obtained by dividing by diagonal elements. If a diagonal element is close to 0, any roundoff error may be magnified in such a division. For example, if a diagonal element should be .00000001, but roundoff error causes it to be calculated as .00000002, division by this number will yield an offdiagonal element that is half of what it should be.
An algorithm is said to be unstable if roundoff error can be magnified in this way or if it can cause the algorithm to fail. The Cholesky algorithm is unstable for singular positive semidefinite matrices h. It is also unstable for positive definite matrices h that have one or more eigenvalues close to 0.
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This glossary post was last updated: 30th December, 2021  0 Views.