This is a very subtle sample problem from ISI MStat PSB 2005 Problem 3. Given that one knows the property of orthogonal matrices its just a counting problem. Give it a thought!
Let be a
orthogonal matrix, where
is even and suppose
, where
denotes the determinant of
. Show that
, where
denotes the
identity matrix.
Orthogonal Matrix
Eigenvalues
Characteristic Polynomial
This is a very simple problem, when you are aware of the basic facts.
We, know that, the eigenvalues of a orthogonal matrix is and
.(
and
if its skew-symmetric). But this given matrix
is not skew-symmetric.(Why??).So let for the matrix
, the algebraic multiplicity of
and
be
and
, respectively.
So, since , hence the algebraic multiplicity of
is definitely odd, since we know by the property of eigenvalues determinant of a matrix is just the product of its eigenvalues.
Now since, is even and the algebraic multiplicity of
i.e.
is odd, hence
is also odd and
.
Hence, the Characteristic Polynomial of , is
, where
is the eigenvalue of
, and in this problem
or
.
Hence, putting , we conclude that,
. Hence we are done !!
Now, suppose is any non-singular matrix, such that
. What can you say about the column space of
?
Keep thinking !!
This is a very subtle sample problem from ISI MStat PSB 2005 Problem 3. Given that one knows the property of orthogonal matrices its just a counting problem. Give it a thought!
Let be a
orthogonal matrix, where
is even and suppose
, where
denotes the determinant of
. Show that
, where
denotes the
identity matrix.
Orthogonal Matrix
Eigenvalues
Characteristic Polynomial
This is a very simple problem, when you are aware of the basic facts.
We, know that, the eigenvalues of a orthogonal matrix is and
.(
and
if its skew-symmetric). But this given matrix
is not skew-symmetric.(Why??).So let for the matrix
, the algebraic multiplicity of
and
be
and
, respectively.
So, since , hence the algebraic multiplicity of
is definitely odd, since we know by the property of eigenvalues determinant of a matrix is just the product of its eigenvalues.
Now since, is even and the algebraic multiplicity of
i.e.
is odd, hence
is also odd and
.
Hence, the Characteristic Polynomial of , is
, where
is the eigenvalue of
, and in this problem
or
.
Hence, putting , we conclude that,
. Hence we are done !!
Now, suppose is any non-singular matrix, such that
. What can you say about the column space of
?
Keep thinking !!