Линейная алгебра: свойства матриц
Ниже перечислены свойства стандартных распространенных типов матриц: идемпотентной, нильпотентной, обратимой, ортогональной, диагональной.
Нильпотентная матрица
Это квадратная матрица A, для которой справедливо An = 0 для n>0.
Некоторые основные свойства нильпотентной матрицы:
- The nilpotent matrix is a square matrix of order n × n.
- The index of a nilpotent matrix having an order of n ×n is either n or a value lesser than n.
- All the eigenvalues of a nilpotent matrix are equal to zero.
- The determinant or the trace of a nilpotent matrix is always zero.
- The nilpotent matrix is a scalar matrix.
- The nilpotent matrix is non-invertible.
Идемпотентная матрица
Это квадратная матрица A, для которой справедливо A2 = A.
Некоторые основные свойства идемпотентной матрицы:
- The idempotent matrix is a square matrix.
- The idempotent matrix is a singular matrix
- The non-diagonal elements can be non-zero elements.
- The eigenvalues of an idempotent matrix is either 0 or 1.
- The trace of an idempotent matrix is equal to the rank of a matrix.
- The trace of an idempotent matrix is always an integer.
Невырожденная/обратимая (non-singular/invertible) матрица
Это квадратная матрица A, для которой справедливо det(A) ≠ 0.
Некоторые основные свойства невырожденной матрицы:
- The non-singular matrix is also called an invertible matrix because its determinant can be computed.
- The product of two non-singular matrices is a non-singular matrix.
- If A is a non-singular matrix, k is a constant, then kA is also a non-singular matrix.
Ортогональная матрица
Это квадратная матрица A, для которой справедливо A · AT = I. I - единичная матрица.
Некоторые основные свойства ортогональной матрицы:
- The product of A and its transpose is an identity matrix: A · AT = AT · A = I.
- Transpose and Inverse are equal: A-1 = AT.
- Determinant is det(A) = ±1. Thus, an orthogonal matrix is always non-singular (as its determinant is NOT 0).
- A diagonal matrix with elements to be 1 or -1 is always orthogonal.
- AT is also orthogonal. Since A-1 = AT, A-1 is also orthogonal.
- The eigenvalues of A are ±1 and the eigenvectors are orthogonal.
- An identity matrix (I) is orthogonal as I · I = I · I = I.
- All the orthogonal matrices are symmetric.
- The product of two orthogonal matrices will also be an orthogonal matrix.
Диагональная матрица
Это квадратная матрица A, недиагональные элементы которой равны 0.
Некоторые основные свойства диагональной матрицы:
- The sum of two diagonal matrices is a diagonal matrix.
- The product of two diagonal matrices (of the same order) is a diagonal matrix where the elements of its principal diagonal are the products of the corresponding elements of the original matrices.
- A + B = B + A
- AB = BA
- Symmetric: AT = A.
- The determinant of a diagonal matrix is the product of its diagonal elements.
- Thus, a diagonal matrix is a non-singular matrix (whose determinant is not zero) only if all of its principal diagonal elements are non-zeros.
- The inverse of a diagonal matrix is a diagonal matrix where the elements of the principal diagonal are the reciprocals of the corresponding elements of the original matrix.
- Any square matrix A can be written as the product A = XDX-1, where D is a diagonal matrix that is formed by the eigenvalues of A and X is formed by the corresponding eigenvectors of A.
Источник: https://www.cuemath.com/algebra/non-singular-matrix/
https://www.toppr.com/guides/maths/matrices/orthogonal-matrix/
18.01.2022