Square matrix

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 Shear mapping or rotation. For example, if $R$ is a square matrix representing a rotation (rotation matrix) and $\backslash mathbf\{v\}$ is a column vector describing the position of a point in space, the product $R\backslash mathbf\{v\}$ yields another column vector describing the position of that point after that rotation. If $\backslash mathbf\{v\}$ is a row vector, the same transformation can be obtained using where $R^\{\backslash mathsf\; T\}$ is the transpose of

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Main diagonal The entries $a\_\{ii\}$ () form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , .

The diagonal of a square matrix from the top right to the bottom left corner is called antidiagonal or counterdiagonal.

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Special kinds

$$ \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix}

$$ \begin{bmatrix} a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & a_{33} \end{bmatrix}

$$ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{bmatrix}

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Diagonal or triangular matrix If all entries outside the main diagonal are zero, $A$ is called a diagonal matrix. If all entries below (resp. above) the main diagonal are zero, $A$ is called an upper (resp. lower) triangular matrix.

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Identity matrix The identity matrix $I\_n$ of size $n$ is the $n\; \backslash times\; n$ matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g. $$I\_1\; =\; \backslash begin\{bmatrix\}\; 1\; \backslash end\{bmatrix\}\; ,\backslash \; I\_2\; =\; \backslash begin\{bmatrix\}$$

        1 & 0 \\
        0 & 1
     \end{bmatrix}
     

,\ \ldots ,\ I_n = \begin{bmatrix}

        1 & 0 & \cdots & 0 \\
        0 & 1 & \cdots & 0 \\
        \vdots & \vdots & \ddots & \vdots \\
        0 & 0 & \cdots & 1
     \end{bmatrix}.
     

It is a square matrix of order and also a special kind of diagonal matrix. The term identity matrix refers to the property of matrix multiplication that $$I\_m\; A\; =\; A\; I\_n\; =\; A$$ for any $m\; \backslash times\; n$ matrix

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Invertible matrix and its inverse A square matrix $A$ is called invertible or non-singular if there exists a matrix $B$ such that $$AB\; =\; BA\; =\; I\_n.$$ If $B$ exists, it is unique and is called the inverse matrix of denoted

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Symmetric or skew-symmetric matrix A square matrix $A$ that is equal to its transpose, i.e., is a symmetric matrix. If instead then $A$ is called a skew-symmetric matrix.

For a complex square matrix often the appropriate analogue of the transpose is the conjugate transpose defined as the transpose of the complex conjugate of A complex square matrix $A$ satisfying $A^*=A$ is called a Hermitian matrix. If instead then $A$ is called a skew-Hermitian matrix.

By the spectral theorem, real symmetric (or complex Hermitian) matrices have an orthogonal (or unitary) eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.

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Definite matrix

$\backslash begin\{bmatrix\}$ 1/4 & 0 \\ 0 & 1 \\ \end{bmatrix}	$\backslash begin\{bmatrix\}$ 1/4 & 0 \\ 0 & -1/4 \end{bmatrix}
Points such that (Ellipse).	Points such that (Hyperbola).

A symmetric -matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors $x\; \backslash in\; \backslash mathbb\{R\}^n$ the associated quadratic form given by $$Q(\backslash mathbf\{x\})\; =\; \backslash mathbf\{x\}^\backslash mathsf\{T\}\; A\; \backslash mathbf\{x\}$$ takes only positive values (respectively only negative values; both some negative and some positive values). If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive. The table at the right shows two possibilities for 2×2 matrices.

Allowing as input two different vectors instead yields the bilinear form associated to : $$B\_A(\backslash mathbf\{x\},\; \backslash mathbf\{y\})\; =\; \backslash mathbf\{x\}^\backslash mathsf\{T\}\; A\; \backslash mathbf\{y\}.$$

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Orthogonal matrix An orthogonal matrix is a square matrix with real number entries whose columns and rows are orthogonal (i.e., orthonormality vectors). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse matrix: $$A^\backslash textsf\{T\}\; =\; A^\{-1\},$$ which entails $$A^\backslash textsf\{T\}\; A\; =\; A\; A^\backslash textsf\{T\}\; =\; I,$$ where I is the identity matrix.

An orthogonal matrix is necessarily invertible (with inverse ), Unitary matrix (), and Normal matrix (). The determinant of any orthogonal matrix is either +1 or −1. The special orthogonal group $\backslash operatorname\{SO\}(n)$ consists of the orthogonal matrices with determinant +1.

The complex number analogue of an orthogonal matrix is a unitary matrix.

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Normal matrix A real or complex square matrix $A$ is called normal matrix if If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian, or unitary, then it is normal. Normal matrices are of interest mainly because they include the types of matrices just listed and form the broadest class of matrices for which the spectral theorem holds.Artin, Algebra, 2nd edition, Pearson, 2018, section 8.6.

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Operations

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Trace The trace, tr( A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors: $$\backslash operatorname\{tr\}(AB)\; =\; \backslash operatorname\{tr\}(BA).$$ This is immediate from the definition of matrix multiplication: $$\backslash operatorname\{tr\}(AB)\; =\; \backslash sum\_\{i=1\}^m\; \backslash sum\_\{j=1\}^n\; A\_\{ij\}\; B\_\{ji\}\; =\; \backslash operatorname\{tr\}(BA).$$ Also, the trace of a matrix is equal to that of its transpose, i.e., $$\backslash operatorname\{tr\}(A)\; =\; \backslash operatorname\{tr\}(A^\{\backslash mathrm\; T\}).$$

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Determinant The determinant $\backslash det(A)$ or $|A|$ of a square matrix $A$ is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in $\backslash mathbb\{R\}^2$) or volume (in $\backslash mathbb\{R\}^3$) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2×2 matrices is given by The determinant of 3×3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalizes these two formulae to all dimensions.

The determinant of a product of square matrices equals the product of their determinants: $$\backslash det(AB)\; =\; \backslash det(A)\; \backslash cdot\; \backslash det(B)$$ Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1×1 matrix, which is its unique entry, or even the determinant of a 0×0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.

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Eigenvalues and eigenvectors A number and a non-zero vector $\backslash mathbf\{v\}$ satisfying $$A\; \backslash mathbf\{v\}\; =\; \backslash lambda\; \backslash mathbf\{v\}$$ are called an eigenvalue and an eigenvector of respectively. Eigen means "own" in German language and in Dutch language. The number is an eigenvalue of an -matrix if and only if is not invertible, which is equivalent to $$\backslash det(A-\backslash lambda\; I)\; =\; 0.$$ The polynomial in an indeterminate given by evaluation of the determinant is called the characteristic polynomial of . It is a monic polynomial of degree n. Therefore the polynomial equation has at most n different solutions, i.e., eigenvalues of the matrix. They may be complex even if the entries of are real. According to the Cayley–Hamilton theorem, , that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.

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See also

• Cartan matrix
• Cayley-Hamilton theorem

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Notes

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External links

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