One application of eigenvalue decomposition is SVD decomposition, which has many applications.
Transmit diversity, channel equalization, blind signal separation.
2. The singular value is the square root of the eigenvalue of a * ah (the * * * yoke transposition of a).
Singular values are often used in decomposition and adaptive filtering of correlation matrices.
* * * yoke matrix
* * * Yoke matrix is also called Hermite matrix. Every element in row I and column J of Hermite matrix is equal to the * * yoke of the elements in row J and column I, and Hermite matrix (or self-* * yoke matrix) is symmetrical with respect to its main diagonal in the form of complex * * * yoke, that is, ai, j = a * j, i.
about
& lt Mathematics & gta = \ \} \ In C < /math & gt；
There are:
& lt Math & gta _ = \ overline} & lt/math & gt；; , in which \ overline & lt/math & gt；; For * * * yoke operators.
Remember to do:
& lt Mathematics & gta = a^h \ quad & lt；; /math & gt；
For example:
& lt Math & gt\ Start
3 & amp2+I \ \ 2-I & amp； 1 \ end & lt； /math & gt；
This is an Hermite array.
Obviously, the elements on the main diagonal of Hermite matrix must be real numbers. For a matrix containing only real elements (real matrix), if it is a symmetric matrix, that is, all elements are symmetric about the main diagonal, then it is also an Hermite matrix. In other words, a real symmetric matrix is a special case of Hermite matrix.
If A and B are Hermite matrices, then their sum A+B is also Hermite matrices; Only when A and B satisfy commutativity (that is, AB = BA), their product is Hermite matrix.
The inverse matrix A- 1 of reversible Hermite matrix A is still Hermite matrix.
If a is Hermite matrix, An is Hermite matrix of positive integer n.
Sum of transposed square matrix c and its yoke
The difference between square matrix C and its yoke transposition
Any square matrix c can be represented by the sum of Hermite matrix a and oblique Hermite matrix b:
& lt Mathematics & GTC = A+B \ quad \ mbox \ quada = \ frac (c+c *) \ quad \ mbox \ quadb = \ frac (c-c *).</Math >
Hermite matrix is a normal matrix, so Hermite matrix can be unitary diagonalized, and the elements of the obtained diagonal matrix are all real numbers. This means that the eigenvalues of Hermite matrix are all real, and the eigenvectors corresponding to different eigenvalues are orthogonal to each other, so we can find a set of orthogonal bases of Cn from these eigenvectors.
The elements of Hermite square matrix of order N constitute a real vector space with dimension n2, because the elements on the main diagonal have one degree of freedom, while the elements above the main diagonal have two degrees of freedom.
If the eigenvalues of Hermite matrix are all positive numbers, then this matrix is positive definite; If they are non-negative, then this matrix is semi-positive.
Hermite sequence (or Hermite vector) refers to the sequence ak (where k = 0, 1, …, n) that satisfies the following conditions:
& lt Math & gt \ im (a _ 0) = 0 \ quad \ mbox \ quad _ k = \ overpass \ quad \ mbox k =1,2 \ dots, n. & lt/math & gt；;
If n is an even number, /2 is a real number.
The discrete Fourier transform of real number sequence is Hermite sequence. On the contrary, the inverse discrete Fourier transform of Hermite sequence is a real number sequence.
* * * Yoke equation: the friction of complex numbers is equal, and the imaginary numbers are reciprocal.