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math_major_111
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Are All symmetric matrices with real number entires Hermitian? What about the other way around-are all Hermitian matrices symmetric?
Penemonie said:Are All symmetric matrices with real number entires Hermitian?
Penemonie said:What about the other way around-are all Hermitian matrices symmetric?
fresh_42 said:What is a Hermitian matrix, and what does this mean, if all entries were real?
A Hermitian matrix is a square matrix that is equal to its own conjugate transpose, meaning that the elements above the main diagonal are the complex conjugates of the elements below the main diagonal. A symmetric matrix, on the other hand, is a square matrix that is equal to its own transpose, meaning that the elements above the main diagonal are the same as the elements below the main diagonal. In other words, a Hermitian matrix is a complex version of a symmetric matrix.
A Hermitian matrix is a special case of a symmetric matrix when the elements of the matrix are real numbers. This means that all Hermitian matrices are also symmetric matrices, but not all symmetric matrices are Hermitian matrices.
Both Hermitian and symmetric matrices are square matrices, meaning they have the same number of rows and columns. They are also both symmetric about the main diagonal, meaning that the elements above and below the main diagonal are reflections of each other. Additionally, both types of matrices have real eigenvalues and orthogonal eigenvectors.
Hermitian and symmetric matrices have numerous applications in mathematics and science. For example, they are used in quantum mechanics to represent observables and physical quantities. They are also used in statistics and data analysis, as well as in computer science for solving systems of linear equations and optimization problems.
No, Hermitian and symmetric matrices cannot be used interchangeably. While they share some properties and similarities, they are distinct types of matrices with different definitions and properties. It is important to use the appropriate terminology when working with these matrices in order to avoid confusion and errors in calculations.