A square matrix of order 4. The entries form the main diagonal of a square matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a11 = 9, a22 = 11, a33 = 4, a44 = 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 . Any two square matrices of the same order can be added and multiplied.
Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if is a square matrix representing a rotation (rotation matrix) and is a column vector describing the position of a point in space, the product yields another column vector describing the position of that point after that rotation. If is a row vector, the same transformation can be obtained using , where is the transpose of .
In mathematics, a squarematrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a squarematrix of order n {\displaystyle...
the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product...
If A is an m × n matrix, then AT is an n × m matrix. In the case of square matrices, AT may also denote the Tth power of the matrix A. For avoiding a...
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real squarematrix whose columns and rows are orthonormal vectors. One way to express...
In linear algebra, the identity matrix of size n {\displaystyle n} is the n × n {\displaystyle n\times n} squarematrix with ones on the main diagonal...
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex squarematrix that is equal to its own conjugate transpose—that is, the element...
In mathematics, a triangular matrix is a special kind of squarematrix. A squarematrix is called lower triangular if all the entries above the main diagonal...
columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number...
is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input...
In linear algebra, an invertible complex squarematrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U...
computer science, an adjacency matrix is a squarematrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices...
In linear algebra, a symmetric matrix is a squarematrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{...
In mathematics, a stochastic matrix is a squarematrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number...
linear algebra, a squarematrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there...
algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements...
covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a squarematrix giving the...
of a squarematrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and...
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a squarematrix of second-order partial derivatives of a scalar-valued function...
algebra, the characteristic polynomial of a squarematrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots...
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [...
{\displaystyle A} must necessarily be a squarematrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings. Examples of 2 × 2 {\displaystyle...
adjugate of a squarematrix A is the transpose of its cofactor matrix and is denoted by adj(A). It is also occasionally known as adjunct matrix, or "adjoint"...
In mathematics, a complex squarematrix A is normal if it commutes with its conjugate transpose A*: A normal ⟺ A ∗ A = A A ∗ . {\displaystyle A{\text{...
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a squarematrix whose entries are either +1 or −1 and whose...
In linear algebra, a nilpotent matrix is a squarematrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} ...
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