Game Development Reference
These systems yield the solutions
where the scalars a and b are arbitrary nonzero constants.
In general, the eigenvalues of a matrix, given by the roots of its characteristic
polynomial, are complex numbers. This means that the corresponding eigen-
vectors can also have complex entries. A type of matrix that is guaranteed to
have real eigenvalues and therefore real eigenvectors, however, is the symmetric
Definition 3.24. An nn
i and j . That is, a matrix whose entries are symmetric about the main diagonal
is called symmetric.
matrix M is symmetric if and only if
The eigenvalues and eigenvectors of symmetric matrices possess the proper-
ties given by the following two theorems.
Theorem 3.25. The eigenvalues of a symmetric matrix M having real entries
are real numbers.
Proof. Let λ be an eigenvalue of the matrix M , and let V be a corresponding ei-
genvector such th at
MVV . Multiplying both sides of this equation on the left
V gives us
by the row vector
V V ,
where the overbar denotes complex conjugation, which for vectors and matrices
is performed componentwise. Since the product of a complex number abi
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