Define the characteristic polynomial of a matrix.

The characteristic polynomial of a matrix is a polynomial equation obtained by finding the determinant of a matrix.

The characteristic polynomial of a matrix A is defined as det(A - λI), where λ is a scalar and I is the identity matrix of the same size as A. In other words, the characteristic polynomial is obtained by subtracting λ from the diagonal entries of A and finding the determinant of the resulting matrix.

The roots of the characteristic polynomial are the eigenvalues of the matrix A. This is because the determinant of A - λI is zero if and only if A - λI is not invertible, which means that there exists a non-zero vector x such that (A - λI)x = 0. This vector x is called an eigenvector of A corresponding to the eigenvalue λ.

The characteristic polynomial is useful in many applications of linear algebra, such as finding eigenvalues and eigenvectors, diagonalizing matrices, and solving differential equations. It also has important theoretical properties, such as the fact that the sum of the eigenvalues of a matrix is equal to its trace, and the product of the eigenvalues is equal to its determinant.

In summary, the characteristic polynomial of a matrix is a polynomial equation obtained by finding the determinant of a matrix, and its roots are the eigenvalues of the matrix. It is a powerful tool in linear algebra with many applications and theoretical properties.