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The following lists matrices whose entries are subject to certain conditions. Many of them apply to square matrices only, that is matrices with the same number of columns and rows. The main diagonal of a square matrix is the diagonal joining the upper left corner and the lower right one or equivalently the entries a i,i. The other diagonal is ...
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 ...
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 .
If a 2 x 2 real matrix has zero trace, its square is a diagonal matrix. The trace of a 2 × 2 complex matrix is used to classify Möbius transformations. First, the matrix is normalized to make its determinant equal to one. Then, if the square of the trace is 4, the corresponding transformation is parabolic.
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} .
In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is greater than or equal to the sum of the magnitudes of all the other (off-diagonal) entries in that row.
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 ...
Similarity is an equivalence relation on the space of square matrices. Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator: Rank; Characteristic polynomial, and attributes that can be derived from it:
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