Discover Linear Algebra

Example 31.4.1. Evaluating a matrix polynomial.

Consider again the polynomial

from Discovery 31.1. As in Subsection 31.3.1, we rewrite this as

where the indeterminate \(X\) now represents an \(n \times n\) matrix. We have also written \(I_n\) to represent the \(n \times n\) identity matrix so that we can be specific about the size of identity used.

First, let's try substituting the \(2 \times 2\) matrix

into \(p(X)\text{:}\)

Now here's an example of substituting the diagonal \(3 \times 3\) matrix

into \(p(X)\text{.}\) Notice how the result matrix is diagonal with the result of evaluating the polynomial at each diagonal entry down the main diagonal.

Example 31.4.2. Patterns of super- and sub-diagonals in powers of a triangular nilpotent matrix.

Here are two examples, one upper and one lower triangular, of powers of nilpotent matrices. Notice how the main diagonal of zeros marches up the super-diagonals of the upper triangular powers and down the sub-diagonals in the lower triangular powers, until eventually the entire matrix becomes zero. (We have omitted the zeros below the main diagonal in the upper triangular example and above the main diagonal in the lower triangular example to emphasize this pattern.)

Example 31.4.3. Scalar-triangular matrices with \(\lambda = 0\).

A scalar-triangular matrix with \(\lambda = 0\) repeated down the main diagonal is always nilpotent. The \(5 \times 5\) matrices in Example 31.4.2 were both of this form, but it is not necessary to have a super- or sub-diagonal of ones to be nilpotent.

Example 31.4.4. A non-triangular nilpotent matrix.

Matrices can be nilpotent without being scalar-triangular. And while we will see that the size of a nilpotent matrix is the “latest” that its powers become zero (Theorem 31.5.3 in Subsection 31.5.1), it can occur before that too.

In Section 33.5, we will see that this particular nilpotent \(5 \times 5\) matrix becomes zero at the third power instead of just at the fifth power because it has a smaller \(3 \times 3\) nilpotent block “hiding” inside it. (In particular, see Example 33.5.4.)