Square matrix

Arbitrary square matrix

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Determinant example

Square matrix refers to a type of matrix in mathematics that has an equal number of rows and columns. The concept of a square matrix is fundamental in various areas of mathematics, including linear algebra, matrix theory, and numerical analysis, as well as in applied disciplines like engineering, physics, and computer science.

Definition

A matrix is considered square if it has n rows and n columns, thus forming an n × n matrix. The elements of a square matrix are typically denoted as a_ij, where i represents the row number and j represents the column number. The main diagonal of a square matrix consists of elements a_ii, where the row number and column number are equal.

Characteristics

Square matrices have several unique characteristics and properties that distinguish them from other forms of matrices. Some of these include:

• Determinant: Only square matrices have a determinant. The determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible and its eigenvalues.

• Invertibility: A square matrix is invertible if there exists another matrix of the same size, known as the inverse matrix, that when multiplied with the original matrix yields the identity matrix. Not all square matrices are invertible.

• Eigenvalues and eigenvectors: These are concepts that apply only to square matrices. An eigenvalue is a scalar that indicates how much the eigenvector is scaled during the transformation represented by the matrix.

• Identity matrix: This is a special type of square matrix where all the elements on the main diagonal are 1, and all other elements are 0. It acts as the identity element in matrix multiplication, meaning any matrix multiplied by the identity matrix is unchanged.

Types of Square Matrices

Several special types of square matrices are recognized, each with its own unique properties:

• Diagonal matrix: A square matrix where all elements outside the main diagonal are zero.
• Symmetric matrix: A square matrix that is equal to its transpose, meaning a_ij = a_ji for all i and j.
• Skew-symmetric matrix: A square matrix that is equal to the negative of its transpose.
• Orthogonal matrix: A square matrix whose transpose is also its inverse.
• Hermitian matrix: In the context of complex numbers, a square matrix that is equal to its conjugate transpose.

Applications

Square matrices are used in solving systems of linear equations, in eigenvalue problems, and in modeling and solving physical problems. They are also essential in the study of graph theory, where the adjacency matrix of a graph is a square matrix.

See Also

• Matrix
• Linear algebra
• Determinant
• Invertible matrix
• Eigenvalues and eigenvectors