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Complex Matrices Hermitian and Positive-Definite Intuition/Visualisation
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When dealing with $BbbR$ it is easy to visualise matrices as how they 'distort' space - I am wondering if, when dealing with $BbbC$ there is also a geometric way of visualising these matrices, or at least some sort of intuition behind them besides how the linear transformation acts on the basis. Specifically, what is the intuition/geometric visualisation behind a hermitian matrix and a positive definite matrix (e.g. an orthogonal matrix geometrically represents rotation/reflection/etc.). Cheers.
linear-algebra abstract-algebra matrices complex-analysis
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Daniele1234
716214
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When dealing with $BbbR$ it is easy to visualise matrices as how they 'distort' space - I am wondering if, when dealing with $BbbC$ there is also a geometric way of visualising these matrices, or at least some sort of intuition behind them besides how the linear transformation acts on the basis. Specifically, what is the intuition/geometric visualisation behind a hermitian matrix and a positive definite matrix (e.g. an orthogonal matrix geometrically represents rotation/reflection/etc.). Cheers.
linear-algebra abstract-algebra matrices complex-analysis
asked yesterday
Daniele1234
716214
The fact that Hermitian matrices can be diagonalized into a real diagonal matrix by a unitary matrix gives some intuition. In particular, this says there's always an orthogonal basis with real eigenvalues ... So we know our space decomposes into a direct sum of orthogonal subspaces in such a way that the Hermitian operator scales each summand by some real number.
– Lorenzo
yesterday
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up vote
1
down vote
favorite
up vote
1
down vote
favorite
When dealing with $BbbR$ it is easy to visualise matrices as how they 'distort' space - I am wondering if, when dealing with $BbbC$ there is also a geometric way of visualising these matrices, or at least some sort of intuition behind them besides how the linear transformation acts on the basis. Specifically, what is the intuition/geometric visualisation behind a hermitian matrix and a positive definite matrix (e.g. an orthogonal matrix geometrically represents rotation/reflection/etc.). Cheers.
linear-algebra abstract-algebra matrices complex-analysis
asked yesterday
Daniele1234
716214
When dealing with $BbbR$ it is easy to visualise matrices as how they 'distort' space - I am wondering if, when dealing with $BbbC$ there is also a geometric way of visualising these matrices, or at least some sort of intuition behind them besides how the linear transformation acts on the basis. Specifically, what is the intuition/geometric visualisation behind a hermitian matrix and a positive definite matrix (e.g. an orthogonal matrix geometrically represents rotation/reflection/etc.). Cheers.
linear-algebra abstract-algebra matrices complex-analysis
asked yesterday
Daniele1234
716214
asked yesterday
Daniele1234
716214
asked yesterday
Daniele1234
716214
asked yesterday
Daniele1234
716214
716214
The fact that Hermitian matrices can be diagonalized into a real diagonal matrix by a unitary matrix gives some intuition. In particular, this says there's always an orthogonal basis with real eigenvalues ... So we know our space decomposes into a direct sum of orthogonal subspaces in such a way that the Hermitian operator scales each summand by some real number.
– Lorenzo
yesterday
add a comment |Â
The fact that Hermitian matrices can be diagonalized into a real diagonal matrix by a unitary matrix gives some intuition. In particular, this says there's always an orthogonal basis with real eigenvalues ... So we know our space decomposes into a direct sum of orthogonal subspaces in such a way that the Hermitian operator scales each summand by some real number.
– Lorenzo
yesterday
The fact that Hermitian matrices can be diagonalized into a real diagonal matrix by a unitary matrix gives some intuition. In particular, this says there's always an orthogonal basis with real eigenvalues ... So we know our space decomposes into a direct sum of orthogonal subspaces in such a way that the Hermitian operator scales each summand by some real number.
– Lorenzo
yesterday
The fact that Hermitian matrices can be diagonalized into a real diagonal matrix by a unitary matrix gives some intuition. In particular, this says there's always an orthogonal basis with real eigenvalues ... So we know our space decomposes into a direct sum of orthogonal subspaces in such a way that the Hermitian operator scales each summand by some real number.
– Lorenzo
yesterday
add a comment |Â
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The fact that Hermitian matrices can be diagonalized into a real diagonal matrix by a unitary matrix gives some intuition. In particular, this says there's always an orthogonal basis with real eigenvalues ... So we know our space decomposes into a direct sum of orthogonal subspaces in such a way that the Hermitian operator scales each summand by some real number.
– Lorenzo
yesterday