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  2. Square matrix - Wikipedia

    en.wikipedia.org/wiki/Square_matrix

    The entries form the main diagonal of a square matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a 11 = 9, a 22 = 11, a 33 = 4, a 44 = 10. In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order .

  3. Diagonal matrix - Wikipedia

    en.wikipedia.org/wiki/Diagonal_matrix

    A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix. The following matrix is square diagonal matrix: [] If the entries are real numbers or complex numbers, then it is a normal matrix as well. In the remainder of this article we will consider only square diagonal matrices, and refer to them ...

  4. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    Applicable to: square, hermitian, positive definite matrix Decomposition: =, where is upper triangular with real positive diagonal entries Comment: if the matrix is Hermitian and positive semi-definite, then it has a decomposition of the form = if the diagonal entries of are allowed to be zero

  5. Jordan normal form - Wikipedia

    en.wikipedia.org/wiki/Jordan_normal_form

    The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue. [3] [4] [5] If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has ...

  6. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.

  7. LU decomposition - Wikipedia

    en.wikipedia.org/wiki/LU_decomposition

    If a square, invertible matrix has an LDU (factorization with all diagonal entries of L and U equal to 1), then the factorization is unique. [8] In that case, the LU factorization is also unique if we require that the diagonal of L {\textstyle L} (or U {\textstyle U} ) consists of ones.

  8. Diagonalizable matrix - Wikipedia

    en.wikipedia.org/wiki/Diagonalizable_matrix

    In linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P {\displaystyle P} and a diagonal matrix D {\displaystyle D} such that P − 1 A P = D {\displaystyle P^{-1}AP=D} .

  9. Matrix (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Matrix_(mathematics)

    An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries a ii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix.