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Let
be an inner product space over and let be a linear operator on . If For all
. If
we call a unitary operator. If we call an orthogonal operator. Similarly for matrix
. If If
, we call a unitary matrix If , we call an orthogonal matrix -
In other words Unitary / Orthogonal operators preserve the lengths of vectors in the vector space.
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(Friedberg 6.18) Let
be a finite dimensional inner product space and let be a linear operator on . Then the following are equivalent (see Normal Matrix) . - If
is an orthonormal basis for , then is an orthonormal basis for . - There exists an orthonormal basis
for such that is an orthonormal basis for . for all .
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(Friedberg 6.18.1) Let
be a linear operator on a finite-dimensional real inner product space . has an orthonormal basis of eigenvectors of with corresponding eigenvalues of absolute value if and only if is both self-adjoint and orthogonal. Another way to say this is that all eigenvalues of
lie along the unit circle -
(Friedberg 6.18.2) Let
be a linear operator on a finite dimensional complex inner product space. Then has an orthonormal basis of eigenvectors of with corresponding eigenvalues of absolute value if and only if is unitary. - If
is orthogonal / unitary, then
- If
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and are unitarily / orthogonally equivalent if and only if there exists a unitary / orthogonal matrix such that This is another form of similarity between matrices if we take the fact that
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(Friedberg 6.19) Let
. Then is normal if and only if is unitarily equivalent to a diagonal matrix. -
(Friedberg 6.20) Let
. Then is symmetric if and only if is orthogonally equivalent to a real diagonal matrix. -
(Friedberg 6.21) Schur’s Theorem (Matrix Form). Let
and whose characteristic polynomial splits over - If
then is a unitarily equivalent to a complex upper triangular matrix. - If
then is an orthogonally equivalent to a real upper triangular matrix.
- If
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(Friedberg e6.5.10) Let
be a complex normal or real symmetric matrix with . We have that - Proof:
is unitarily / orthogonally equivalent to a diagonal matrix, that is . The diagonal matrix has the eigenvalues as its entries. We have by virtue of having the same shape as Similarly,
- Proof:
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(Friedberg e6.5.12) Let
be an real symmetric or complex normal matrix. Let . We have the determinant - Proof:
is unitarily / orthogonally equivalent to a diagonal matrix, that is . The diagonal matrix has the eigenvalues as its entries. We have
- Proof: