In mathematics, the Courant minimax principle gives the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of a real
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...
. It is named after
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
.
Introduction
The Courant minimax principle gives a condition for finding the eigenvalues for a real symmetric matrix. The Courant minimax principle is as follows:
For any real symmetric matrix ''A'',
:
where
is any
matrix.
Notice that the vector ''x'' is an
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
to the corresponding eigenvalue ''λ''.
The Courant minimax principle is a result of the maximum theorem, which says that for
, ''A'' being a real symmetric matrix, the largest eigenvalue is given by
, where
is the corresponding eigenvector. Also (in the maximum theorem) subsequent eigenvalues
and eigenvectors
are found by induction and orthogonal to each other; therefore,
with