From A First Course in Linear Algebra
Version 1.06
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/
This Section Under Construction
Our first decomposition applies only to diagonalizable (Definition DZM) matrices, and yields a decomposition into a sum of very simple matrices.
Theorem ROD
Rank One Decomposition
Suppose that is a
diagonalizable matrix of size
and rank . Then
there are square
matrices ,
each of size
and rank
such that
Furthermore, if are the nonzero eigenvalues of , then there are two sets of linearly independent vectors from ,
such that , .
Proof The proof is constructive. Generally, we will diagonalize , creating a nonsingular matrix and a diagonal matrix . Then we split up the diagonal matrix into a sum of matrices with a single nonzero entry (on the diagonal). This fundamentally creates the decomposition in the statement of the theorem, the remainder is just bookkeeping. The vectors in and will result from the columns of and the rows of .
Let be the eigenvalues of (repeated according to their algebraic multiplicity). If has rank , then (Theorem RPNC). The null space of is the eigenspace of the eigenvalue (Theorem EMNS), so it follows that the algebraic multiplicity of is , . Presume that the complete list of eigenvalues is ordered so that for .
Since is hypothesized to be diagonalizable, there exists a diagonal matrix and an invertible matrix , such that . We can rearrange tis equation to read, . Also, the proof of Theorem DC says that the diagonal elements of are the eigenvalues of and we have the flexibility to assume they lie on the diagonal in the same order as we have specified above. Now, let be the columns of , and let be the rows of converted to column vectors. With little motivation other than the statement of the theorem, define size matrices , by . Finally, let be the size matrix that is totally zero, other than having in row and column .
With everything in place, we compute entry-by-entry,
So by Definition ME we have the desired equality of matrices. The careful reader will have noted that , , since in these instances. To get the sets and from and , simply discard the last vectors. We can safely ignore (or remove) from the summation just derived.
One last assertion to check. What is the rank of , ? Every row of is a scalar multiple of , row of the nonsingular matrix (Theorem MIMI). As a row of a nonsingular matrix, cannot be all zeros. In particular, row of is obtained as a scalar multiple of by the scalar . We have restricted ourselves to the nonzero eigenvalues of , and as is nonsingular, some entry of is nonzero. This all implies that some row of will be nonzero. Now consider row-reducing . Swap the nonzero row up into row 1. Use scalar multiples of this row to zero out every other row. This leaves a single nonzero row in the reduced row-echelon form, so has rank one.
We record two observations that was not stated in our theorem above. First, the vectors in , chosen as columns of , are eigenvectors of . Second, the product of two vectors from and in the opposite order, by which we mean , is the entry in row and column of the matrix product (Theorem EMP). In particular,
We give two computational examples. One small, one a bit bigger.
Example ROD2
Rank one decomposition, size 2
Consider the
matrix,
By the techniques of Chapter E we find the eigenvalues and eigenspaces,
With distinct eigenvalues, Theorem DED tells us that is diagonalizable, and with no zero eigenvalues we see that has full rank. Theorem DC says we can construct the nonsingular matrix with eigenvectors of as columns, so we have
From these matrices we obtain the sets of vectors
And we have the matrices,
And you can easily verify that .
Here’s a slightly larger example, and the matrix does not have full rank.
Example ROD4
Rank one decomposition, size 4
Consider the
matrix,
By the techniques of Chapter E we find the eigenvalues and eigenvectors,
The algebraic and geometric multiplicities of each eigenvalue are equal, so Theorem DMFE tells us that is diagonalizable. With a single zero eigenvalue we see that has rank . Theorem DC says we can construct the nonsingular matrix with eigenvectors of as columns, so we have
Since , we need only collect three vectors from each of these matrices,
And we obtain the matrices,
Then we verify that