7.2. THE CAYLEY HAMILTON THEOREM∗ 139

By the formula for the expansion of a determinant along a column,

xi =1

det(A)det

∗ · · · y1 · · · ∗...

......

∗ · · · yn · · · ∗

 ,

where here the ith column of A is replaced with the column vector (y1 · · · ·,yn)T , and the

determinant of this modified matrix is taken and divided by det(A). This formula is knownas Cramer’s rule.

7.1.10 Upper Triangular MatricesDefinition 7.1.16 A matrix M, is upper triangular if Mi j = 0 whenever i > j. Thus such amatrix equals zero below the main diagonal, the entries of the form Mii as shown.

∗ ∗ · · · ∗

0 ∗. . .

......

. . .. . . ∗

0 · · · 0 ∗

A lower triangular matrix is defined similarly as a matrix for which all entries above themain diagonal are equal to zero.

With this definition, here is a simple corollary of Theorem 7.1.13.

Corollary 7.1.17 Let M be an upper (lower) triangular matrix. Then det(M) is obtainedby taking the product of the entries on the main diagonal.

7.2 The Cayley Hamilton Theorem∗

Definition 7.2.1 Let A be an n×n matrix. The characteristic polynomial is defined as

qA (t)≡ det(tI−A)

and the solutions to qA (t) = 0 are called eigenvalues. For A a matrix and p(t) = tn +an−1tn−1 + · · ·+a1t +a0, denote by p(A) the matrix defined by

p(A)≡ An +an−1An−1 + · · ·+a1A+a0I.

The explanation for the last term is that A0 is interpreted as I, the identity matrix.

The Cayley Hamilton theorem states that every matrix satisfies its characteristic equa-tion, that equation defined by qA (t) = 0. It is one of the most important theorems in linearalgebra1. The proof in this section is not the most general proof, but works well when thefield of scalars is R or C. The following lemma will help with its proof.

1A special case was first proved by Hamilton in 1853. The general case was announced by Cayley some timelater and a proof was given by Frobenius in 1878.