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:

  1. The nilpotent matrix is a square matrix of order n × n.
  2. The index of a nilpotent matrix having an order of n ×n is either n or a value lesser than n.
  3. All the eigenvalues of a nilpotent matrix are equal to zero.
  4. The determinant or the trace of a nilpotent matrix is always zero.
  5. The nilpotent matrix is a scalar matrix.
  6. The nilpotent matrix is non-invertible.

Идемпотентная матрица

Это квадратная матрица A, для которой справедливо A2 = A.

Некоторые основные свойства идемпотентной матрицы:

  1. The idempotent matrix is a square matrix.
  2. The idempotent matrix is a singular matrix
  3. The non-diagonal elements can be non-zero elements.
  4. The eigenvalues of an idempotent matrix is either 0 or 1.
  5. The trace of an idempotent matrix is equal to the rank of a matrix.
  6. 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:

  1. The non-singular matrix is also called an invertible matrix because its determinant can be computed.
  2. The product of two non-singular matrices is a non-singular matrix.
  3. 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:

  1. The product of A and its transpose is an identity matrix: A · AT = AT · A = I.
  2. Transpose and Inverse are equal: A-1 = AT.
  3. Determinant is det(A) = ±1. Thus, an orthogonal matrix is always non-singular (as its determinant is NOT 0).
  4. A diagonal matrix with elements to be 1 or -1 is always orthogonal.
  5. AT is also orthogonal. Since A-1 = AT, A-1 is also orthogonal.
  6. The eigenvalues of A are ±1 and the eigenvectors are orthogonal.
  7. An identity matrix (I) is orthogonal as I · I = I · I = I.
  8. All the orthogonal matrices are symmetric.
  9. 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:

  1. The sum of two diagonal matrices is a diagonal matrix.
  2. 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.
  3. A + B = B + A
  4. AB = BA
  5. Symmetric: AT = A.
  6. The determinant of a diagonal matrix is the product of its diagonal elements.
  7. 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.
  8. 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.
  9. 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


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