Matrix Operations - Revision history

Khanh: /* Eigenvalues and eigenvectors */

2022-01-10T22:05:55Z

Eigenvalues and eigenvectors

Khanh: /* Main operations */

2022-01-10T22:00:32Z

Main operations

Khanh: /* Orthogonal matrix */

2022-01-10T21:56:57Z

Orthogonal matrix

Khanh: /* Definite matrix */

2022-01-10T21:39:52Z

Definite matrix

Khanh: /* Definite matrix */

2022-01-10T21:39:10Z

Definite matrix

Khanh: /* Square matrix */

2022-01-10T21:31:33Z

Square matrix

Khanh: /* Identity matrix */

2022-01-10T21:30:50Z

Identity matrix

Khanh: /* Main types */

2022-01-10T21:24:02Z

Main types

Khanh: /* Square matrix */

2022-01-10T21:16:42Z

Square matrix

Khanh: /* Linear transformations */

2022-01-10T21:12:28Z

Linear transformations

@@ Line 373: / Line 373: @@
 are called an ''eigenvalue'' and an ''eigenvector'' of '''A''', respectively. The number λ is an eigenvalue of an ''n''×''n''-matrix '''A''' if and only if '''A'''−λ'''I'''<sub>''n''</sub> is not invertible, which is equivalent to
 :<math>\det(\mathbf{A}-\lambda \mathbf{I}) = 0.</math>
-The polynomial ''p''{{sub|'''A'''}} in an indeterminate ''X'' given by evaluation of the determinant det(''X'''''I'''<sub>''n''</sub>−'''A''') is called the characteristic polynomial of '''A'''. It is a monic polynomial of degree ''n''. Therefore the polynomial equation ''p''<sub>'''A'''</sub>(λ) = 0 has at most ''n'' different solutions, that is, eigenvalues of the matrix. They may be complex even if the entries of '''A''' are real. According to the Cayley–Hamilton theorem, ''p''<sub>'''A'''</sub>('''A''') = '''0''', that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.
+The polynomial ''p''<sub>'''A'''</sub> in an indeterminate ''X'' given by evaluation of the determinant det(''X'''''I'''<sub>''n''</sub>−'''A''') is called the characteristic polynomial of '''A'''. It is a monic polynomial of degree ''n''. Therefore the polynomial equation ''p''<sub>'''A'''</sub>(λ) = 0 has at most ''n'' different solutions, that is, eigenvalues of the matrix. They may be complex even if the entries of '''A''' are real. According to the Cayley–Hamilton theorem, ''p''<sub>'''A'''</sub>('''A''') = '''0''', that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.
 == Resources ==

@@ Line 353: / Line 353: @@
 :<math>\operatorname{tr}(\mathbf{AB}) = \sum_{i=1}^m \sum_{j=1}^n a_{ij} b_{ji} = \operatorname{tr}(\mathbf{BA}).</math>
 It follows that the trace of the product of more than two matrices is independent of cyclic permutations of the matrices, however this does not in general apply for arbitrary permutations (for example, tr('''ABC''') ≠ tr('''BAC'''), in general). Also, the trace of a matrix is equal to that of its transpose, that is,
-:tr('''A''') = tr('''A'''{{sup|T}}).
+:tr('''A''') = tr('''A'''<sup>T</sup>).
 ====Determinant====
-[[File:Determinant example.svg|thumb|300px|right|A linear transformation on '''R'''{{sup|2}} given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.]]
+[[File:Determinant example.svg|thumb|300px|right|A linear transformation on '''R'''<sup>2</sup> given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.]]
-The ''determinant'' of a square matrix '''A''' (denoted det('''A''') or |'''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 '''R'''{{sup|2}}) or volume (in '''R'''{{sup|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 a square matrix '''A''' (denoted det('''A''') or |'''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 '''R'''<sup>2</sup>) or volume (in '''R'''<sup>3</sup>) 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-by-2 matrices is given by
@@ Line 366: / Line 366: @@
 The determinant of a product of square matrices equals the product of their determinants:
 :det('''AB''') = det('''A''') · 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, that is, determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems 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.
+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, that is, determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems 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.
 ====Eigenvalues and eigenvectors====
 A number λ and a non-zero vector '''v''' satisfying
 :<math>Av = \lambda v</math>
-are called an ''eigenvalue'' and an ''eigenvector'' of '''A''', respectively. The number λ is an eigenvalue of an ''n''×''n''-matrix '''A''' if and only if '''A'''−λ'''I'''{{sub|''n''}} is not invertible, which is equivalent to
+are called an ''eigenvalue'' and an ''eigenvector'' of '''A''', respectively. The number λ is an eigenvalue of an ''n''×''n''-matrix '''A''' if and only if '''A'''−λ'''I'''<sub>''n''</sub> is not invertible, which is equivalent to
 :<math>\det(\mathbf{A}-\lambda \mathbf{I}) = 0.</math>
-The polynomial ''p''{{sub|'''A'''}} in an indeterminate ''X'' given by evaluation of the determinant det(''X'''''I'''{{sub|''n''}}−'''A''') is called the [[characteristic polynomial]] of '''A'''. It is a monic polynomial of degree ''n''. Therefore the polynomial equation ''p''{{sub|'''A'''}}(λ){{nbsp}}={{nbsp}}0 has at most ''n'' different solutions, that is, eigenvalues of the matrix. They may be complex even if the entries of '''A''' are real. According to the Cayley–Hamilton theorem, ''p''{{sub|'''A'''}}('''A''') = '''0''', that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.
+The polynomial ''p''{{sub|'''A'''}} in an indeterminate ''X'' given by evaluation of the determinant det(''X'''''I'''<sub>''n''</sub>−'''A''') is called the characteristic polynomial of '''A'''. It is a monic polynomial of degree ''n''. Therefore the polynomial equation ''p''<sub>'''A'''</sub>(λ) = 0 has at most ''n'' different solutions, that is, eigenvalues of the matrix. They may be complex even if the entries of '''A''' are real. According to the Cayley–Hamilton theorem, ''p''<sub>'''A'''</sub>('''A''') = '''0''', that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.
 == Resources ==

@@ Line 339: / Line 339: @@
 which entails
 :<math>\mathbf{A}^\mathrm{T} \mathbf{A} = \mathbf{A} \mathbf{A}^\mathrm{T} = \mathbf{I}_n,</math>
-where '''I'''{{sub|''n''}} is the identity matrix of size ''n''.
+where '''I'''<sub>''n''</sub> is the identity matrix of size ''n''.
-An orthogonal matrix '''A''' is necessarily invertible (with inverse '''A'''{{sup|&minus;1}} = '''A'''{{sup|T}}), unitary ('''A'''{{sup|&minus;1}} = '''A'''*), and normal '''A'''*'''A''' = '''AA'''*). The determinant of any orthogonal matrix is either {{math|+1}} or {{math|−1}}. A ''special orthogonal matrix'' is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant {{math|+1}} is a pure rotation without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant {{math|-1}} reverses the orientation, i.e., is a composition of a pure reflection and a (possibly null) rotation. The identity matrices have determinant {{math|1}}, and are pure rotations by an angle zero.
+An orthogonal matrix '''A''' is necessarily invertible (with inverse '''A'''<sup>-1</sup> = '''A'''<sup>T</sup>), unitary ('''A'''<sup>-1</sup> = '''A'''*), and normal '''A'''*'''A''' = '''AA'''*). The determinant of any orthogonal matrix is either {{math|+1}} or {{math|−1}}. A ''special orthogonal matrix'' is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant {{math|+1}} is a pure rotation without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant {{math|-1}} reverses the orientation, i.e., is a composition of a pure reflection and a (possibly null) rotation. The identity matrices have determinant {{math|1}}, and are pure rotations by an angle zero.
 The complex analogue of an orthogonal matrix is a unitary matrix.

@@ Line 332: / Line 332: @@
 :''B''<sub>'''A'''</sub> ('''x''', '''y''') = '''x'''<sup>T</sup>'''Ay'''.
-In the case of complex matrices, the same terminology and result apply, with ''symmetric matrix'', ''quadratic form'', ''bilinear form'', and ''transpose'' '''x'''<sup>T</sup> replaced respectively by  Hermitian matrix, Hermitian form, sesquilinear form, and conjugate transpose '''x'''<sup>HM/sup>.
+In the case of complex matrices, the same terminology and result apply, with ''symmetric matrix'', ''quadratic form'', ''bilinear form'', and ''transpose'' '''x'''<sup>T</sup> replaced respectively by  Hermitian matrix, Hermitian form, sesquilinear form, and conjugate transpose '''x'''<sup>H</sup>.
 ====Orthogonal matrix====

@@ Line 317: / Line 317: @@
 \end{bmatrix}</math>
 |-
-| ''Q''(''x'', ''y'') = {{sfrac|1|4}} ''x''{{sup|2}} + ''y''{{sup|2}}
+| ''Q''(''x'', ''y'') = <math> \tfrac{1}{4}</math> ''x''<sup>2</sup> + ''y''<sup>2</sup>
-| ''Q''(''x'', ''y'') = {{sfrac|1|4}} ''x''{{sup|2}} − 1/4 ''y''{{sup|2}}
+| ''Q''(''x'', ''y'') = <math> \tfrac{1}{4}</math> ''x''<sup>2</sup> - <math> \tfrac {1}{4}</math> ''y''<sup>2</sup>
 |-
 | [[File:Ellipse in coordinate system with semi-axes labelled.svg|150px]] <br>Points such that ''Q''(''x'',''y'')=1 <br> (Ellipse).
@@ Line 324: / Line 324: @@
 |}
 A symmetric real matrix {{math|'''A'''}} is called ''positive-definite'' if the associated quadratic form
-:<span id="quadratic forms">{{math|1=''f''{{spaces|hair}}('''x''') = '''x'''{{sup|T}}'''A{{nbsp}}x'''}}</span>
+:<span id="quadratic forms">''f''('''x''') = '''x'''<sup>T</sup>'''A '''x'''</span>
-has a positive value for every nonzero vector {{math|'''x'''}} in {{math|'''R'''{{sup|''n''}}}}. If {{math|''f''{{spaces|hair}}('''x''')}} only yields negative values then {{math|'''A'''}} is ''negative-definite''; if {{math|''f''}} does produce both negative and positive values then {{math|'''A'''}} is ''indefinite''. If the quadratic form {{math|''f''}} yields only non-negative values (positive or zero), the symmetric matrix is called ''positive-semidefinite'' (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.
+has a positive value for every nonzero vector '''x''' in '''R'''<sup>''n''</sup>. If ''f''('''x''') only yields negative values then {{math|'''A'''}} is ''negative-definite''; if {{math|''f''}} does produce both negative and positive values then {{math|'''A'''}} is ''indefinite''. If the quadratic form {{math|''f''}} yields only non-negative values (positive or zero), the symmetric matrix is called ''positive-semidefinite'' (or if only non-positive values, then 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, that is, the matrix is positive-semidefinite and it is invertible. The table at the right shows two possibilities for 2-by-2 matrices.
 Allowing as input two different vectors instead yields the bilinear form associated to {{math|'''A'''}}:
-:{{math|1=''B''{{sub|'''A'''}} ('''x''', '''y''') = '''x'''{{sup|T}}'''Ay'''}}.
+:''B''<sub>'''A'''</sub> ('''x''', '''y''') = '''x'''<sup>T</sup>'''Ay'''.
-In the case of complex matrices, the same terminology and result apply, with ''symmetric matrix'', ''quadratic form'', ''bilinear form'', and ''transpose'' {{math|'''x'''{{sup|T}}}} replaced respectively by  Hermitian matrix, Hermitian form, sesquilinear form, and conjugate transpose {{math|'''x'''{{sup|H}}}}.
+In the case of complex matrices, the same terminology and result apply, with ''symmetric matrix'', ''quadratic form'', ''bilinear form'', and ''transpose'' '''x'''<sup>T</sup> replaced respectively by  Hermitian matrix, Hermitian form, sesquilinear form, and conjugate transpose '''x'''<sup>HM/sup>.
 ====Orthogonal matrix====

@@ Line 236: / Line 236: @@
 ==Square matrix==
-A square matrix is a matrix with the same number of rows and columns.<ref name=":4" /> 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.
+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 ''n.'' Any two square matrices of the same order can be added and multiplied.
 The entries ''a''<sub>''ii''</sub> 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.

@@ Line 273: / Line 273: @@
 ====Identity matrix====
-The ''identity matrix'' '''I'''{{sub|''n''}} of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example,
+The ''identity matrix'' '''I'''<sub>''n''</sub> of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example,
 :<math>
 \mathbf{I}_1 = \begin{bmatrix} 1 \end{bmatrix},

@@ Line 288: / Line 288: @@
 \end{bmatrix}
 </math>
-It is a square matrix of order ''n'', and also a special kind of [[diagonal matrix]]. It is called an identity matrix because multiplication with it leaves a matrix unchanged:
+It is a square matrix of order ''n'', and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:
-:'''AI'''{{sub|''n''}} = '''I'''{{sub|''m''}}'''A''' = '''A''' for any ''m''-by-''n'' matrix '''A'''.
+:'''AI'''<sub>''n''</sub> = '''I'''<sub>''m''</sub>'''A''' = '''A''' for any ''m''-by-''n'' matrix '''A'''.
 A nonzero scalar multiple of an identity matrix is called a ''scalar'' matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.
 ====Symmetric or skew-symmetric matrix====
-A square matrix '''A''' that is equal to its transpose, that is, '''A''' = '''A'''{{sup|T}}, is a [[symmetric matrix]]. If instead, '''A''' is equal to the negative of its transpose, that is, '''A''' = −'''A'''{{sup|T}}, then '''A''' is a [[skew-symmetric matrix]]. In complex matrices, symmetry is often replaced by the concept of [[Hermitian matrix|Hermitian matrices]], which satisfy '''A'''{{sup|∗}} = '''A''', where the star or [[asterisk]] denotes the [[conjugate transpose]] of the matrix, that is, the transpose of the [[complex conjugate]] of '''A'''.
+A square matrix '''A''' that is equal to its transpose, that is, '''A''' = '''A'''<sup>T</sup>, is a symmetric matrix. If instead, '''A''' is equal to the negative of its transpose, that is, '''A''' = −'''A'''<sup>T</sup>, then '''A''' is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy '''A'''<sup>∗</sup> = '''A''', where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of '''A'''.
-By the [[spectral theorem]], real symmetric matrices and complex Hermitian matrices have an [[eigenbasis]]; that is, every vector is expressible as a [[linear combination]] of eigenvectors. In both cases, all eigenvalues are real. This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see [[#Infinite matrices|below]].
+By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns.
 ====Invertible matrix and its inverse====
-A square matrix '''A''' is called ''[[invertible matrix|invertible]]'' or ''non-singular'' if there exists a matrix '''B''' such that
+A square matrix '''A''' is called ''invertible'' or ''non-singular'' if there exists a matrix '''B''' such that
-:'''AB''' = '''BA''' = '''I'''{{sub|''n''}} ,<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.2.28}}</ref><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.5.13}}</ref>
+:'''AB''' = '''BA''' = '''I'''<sub>''n''</sub> ,
-where '''I'''{{sub|''n''}} is the ''n''×''n'' [[identity matrix]] with 1s on the [[main diagonal]] and 0s elsewhere. If '''B''' exists, it is unique and is called the ''[[Invertible matrix|inverse matrix]]'' of '''A''', denoted '''A'''{{sup|−1}}.
+where '''I'''<sub>''n''</sub> is the ''n''×''n'' identity matrix with 1s on the main diagonal and 0s elsewhere. If '''B''' exists, it is unique and is called the ''inverse matrix'' of '''A''', denoted '''A'''<sup>−1</sup>.
 ====Definite matrix====
@@ Line 320: / Line 320: @@
 | ''Q''(''x'', ''y'') = {{sfrac|1|4}} ''x''{{sup|2}} − 1/4 ''y''{{sup|2}}
 |-
-| [[File:Ellipse in coordinate system with semi-axes labelled.svg|150px]] <br>Points such that ''Q''(''x'',''y'')=1 <br> ([[Ellipse]]).
+| [[File:Ellipse in coordinate system with semi-axes labelled.svg|150px]] <br>Points such that ''Q''(''x'',''y'')=1 <br> (Ellipse).
-| [[File:Hyperbola2 SVG.svg|150px]] <br> Points such that ''Q''(''x'',''y'')=1 <br> ([[Hyperbola]]).
+| [[File:Hyperbola2 SVG.svg|150px]] <br> Points such that ''Q''(''x'',''y'')=1 <br> (Hyperbola).
 |}
-A symmetric real matrix {{math|'''A'''}} is called [[positive-definite matrix|''positive-definite'']] if the associated [[quadratic form]]
+A symmetric real matrix {{math|'''A'''}} is called ''positive-definite'' if the associated quadratic form
 :<span id="quadratic forms">{{math|1=''f''{{spaces|hair}}('''x''') = '''x'''{{sup|T}}'''A{{nbsp}}x'''}}</span>
-has a positive value for every nonzero vector {{math|'''x'''}} in {{math|'''R'''{{sup|''n''}}}}. If {{math|''f''{{spaces|hair}}('''x''')}} only yields negative values then {{math|'''A'''}} is [[definiteness of a matrix#Negative definite|''negative-definite'']]; if {{math|''f''}} does produce both negative and positive values then {{math|'''A'''}} is [[definiteness of a matrix#Indefinite|''indefinite'']]. If the quadratic form {{math|''f''}} yields only non-negative values (positive or zero), the symmetric matrix is called ''positive-semidefinite'' (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.
+has a positive value for every nonzero vector {{math|'''x'''}} in {{math|'''R'''{{sup|''n''}}}}. If {{math|''f''{{spaces|hair}}('''x''')}} only yields negative values then {{math|'''A'''}} is ''negative-definite''; if {{math|''f''}} does produce both negative and positive values then {{math|'''A'''}} is ''indefinite''. If the quadratic form {{math|''f''}} yields only non-negative values (positive or zero), the symmetric matrix is called ''positive-semidefinite'' (or if only non-positive values, then 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, that is, the matrix is positive-semidefinite and it is invertible. The table at the right shows two possibilities for 2-by-2 matrices.
-Allowing as input two different vectors instead yields the [[bilinear form]] associated to {{math|'''A'''}}:<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Example 4.0.6, p. 169}}</ref>
+Allowing as input two different vectors instead yields the bilinear form associated to {{math|'''A'''}}:
 :{{math|1=''B''{{sub|'''A'''}} ('''x''', '''y''') = '''x'''{{sup|T}}'''Ay'''}}.
-In the case of complex matrices, the same terminology and result apply, with ''symmetric matrix'', ''quadratic form'', ''bilinear form'', and ''transpose'' {{math|'''x'''{{sup|T}}}} replaced respectively by  [[Hermitian matrix]], [[Hermitian form]], [[sesquilinear form]], and [[conjugate transpose]] {{math|'''x'''{{sup|H}}}}.
+In the case of complex matrices, the same terminology and result apply, with ''symmetric matrix'', ''quadratic form'', ''bilinear form'', and ''transpose'' {{math|'''x'''{{sup|T}}}} replaced respectively by  Hermitian matrix, Hermitian form, sesquilinear form, and conjugate transpose {{math|'''x'''{{sup|H}}}}.
 ====Orthogonal matrix====
-An ''orthogonal matrix'' is a [[#Square matrices|square matrix]] with [[real number|real]] entries whose columns and rows are [[orthogonal]] [[unit vector]]s (that is, [[orthonormality|orthonormal]] vectors). Equivalently, a matrix '''A''' is orthogonal if its [[transpose]] is equal to its [[invertible matrix|inverse]]:
+An ''orthogonal matrix'' is a square matrix with real entries whose columns and rows are orthogonal unit vectors (that is, orthonormal vectors). Equivalently, a matrix '''A''' is orthogonal if its transpose is equal to its inverse:
 :<math>\mathbf{A}^\mathrm{T}=\mathbf{A}^{-1}, \,</math>
 which entails
 :<math>\mathbf{A}^\mathrm{T} \mathbf{A} = \mathbf{A} \mathbf{A}^\mathrm{T} = \mathbf{I}_n,</math>
-where '''I'''{{sub|''n''}} is the [[identity matrix]] of size ''n''.
+where '''I'''{{sub|''n''}} is the identity matrix of size ''n''.
-An orthogonal matrix '''A''' is necessarily [[invertible matrix|invertible]] (with inverse {{nowrap|1='''A'''{{sup|&minus;1}} = '''A'''{{sup|T}}}}), [[unitary matrix|unitary]] ({{nowrap|1='''A'''{{sup|&minus;1}} = '''A'''*}}), and [[normal matrix|normal]] ({{nowrap|1='''A'''*'''A''' = '''AA'''*}}). The [[determinant]] of any orthogonal matrix is either {{math|+1}} or {{math|−1}}. A ''special orthogonal matrix'' is an orthogonal matrix with [[determinant]] +1. As a [[linear transformation]], every orthogonal matrix with determinant {{math|+1}} is a pure [[rotation (mathematics)|rotation]] without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant {{math|-1}} reverses the orientation, i.e., is a composition of a pure [[reflection (mathematics)|reflection]] and a (possibly null) rotation. The identity matrices have determinant {{math|1}}, and are pure rotations by an angle zero.
+An orthogonal matrix '''A''' is necessarily invertible (with inverse '''A'''{{sup|&minus;1}} = '''A'''{{sup|T}}), unitary ('''A'''{{sup|&minus;1}} = '''A'''*), and normal '''A'''*'''A''' = '''AA'''*). The determinant of any orthogonal matrix is either {{math|+1}} or {{math|−1}}. A ''special orthogonal matrix'' is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant {{math|+1}} is a pure rotation without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant {{math|-1}} reverses the orientation, i.e., is a composition of a pure reflection and a (possibly null) rotation. The identity matrices have determinant {{math|1}}, and are pure rotations by an angle zero.
 The complex analogue of an orthogonal matrix is a unitary matrix.

@@ Line 236: / Line 236: @@
 ==Square matrix==
-A [[square matrix]] is a matrix with the same number of rows and columns.<ref name=":4" /> 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.
+A square matrix is a matrix with the same number of rows and columns.<ref name=":4" /> 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''{{sub|''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.
+The entries ''a''<sub>''ii''</sub> 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.
 ===Main types===
@@ Line 244: / Line 244: @@
 ! Name !! Example with ''n'' = 3
 |-
-| [[Diagonal matrix]] || style="text-align:center;" | <math>
+| Diagonal matrix || style="text-align:center;" | <math>
 \begin{bmatrix}
 a_{11} & 0      & 0 \\
@@ Line 252: / Line 252: @@
 </math>
 |-
-| [[Lower triangular matrix]] || style="text-align:center;" | <math>
+| Lower triangular matrix || style="text-align:center;" | <math>
 \begin{bmatrix}
 a_{11} &      0 & 0 \\
@@ Line 260: / Line 260: @@
 </math>
 |-
-| [[Upper triangular matrix]] || style="text-align:center;" | <math>
+| Upper triangular matrix || style="text-align:center;" | <math>
 \begin{bmatrix}
 a_{11} & a_{12} & a_{13} \\
@@ Line 270: / Line 270: @@
 ====Diagonal and triangular matrix====
-If all entries of '''A''' below the main diagonal are zero, '''A''' is called an ''upper [[triangular matrix]]''. Similarly if all entries of ''A'' above the main diagonal are zero, '''A''' is called a ''lower triangular matrix''. If all entries outside the main diagonal are zero, '''A''' is called a [[diagonal matrix]].
+If all entries of '''A''' below the main diagonal are zero, '''A''' is called an ''upper triangular matrix''. Similarly if all entries of ''A'' above the main diagonal are zero, '''A''' is called a ''lower triangular matrix''. If all entries outside the main diagonal are zero, '''A''' is called a diagonal matrix.
 ====Identity matrix====
-The ''identity matrix'' '''I'''{{sub|''n''}} of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the [[main diagonal]] are equal to 1 and all other elements are equal to 0, for example,
+The ''identity matrix'' '''I'''{{sub|''n''}} of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example,
 :<math>
 \mathbf{I}_1 = \begin{bmatrix} 1 \end{bmatrix},
@@ Line 306: / Line 306: @@
 {| class="wikitable" style="float:right; text-align:center; margin:0ex 0ex 2ex 2ex;"
 |-
-! [[Positive definite matrix]] !! [[Indefinite matrix]]
+! Positive definite matrix !! Indefinite matrix
 |-
 | <math>\begin{bmatrix}
@@ Line 343: / Line 343: @@
 An orthogonal matrix '''A''' is necessarily [[invertible matrix|invertible]] (with inverse {{nowrap|1='''A'''{{sup|&minus;1}} = '''A'''{{sup|T}}}}), [[unitary matrix|unitary]] ({{nowrap|1='''A'''{{sup|&minus;1}} = '''A'''*}}), and [[normal matrix|normal]] ({{nowrap|1='''A'''*'''A''' = '''AA'''*}}). The [[determinant]] of any orthogonal matrix is either {{math|+1}} or {{math|−1}}. A ''special orthogonal matrix'' is an orthogonal matrix with [[determinant]] +1. As a [[linear transformation]], every orthogonal matrix with determinant {{math|+1}} is a pure [[rotation (mathematics)|rotation]] without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant {{math|-1}} reverses the orientation, i.e., is a composition of a pure [[reflection (mathematics)|reflection]] and a (possibly null) rotation. The identity matrices have determinant {{math|1}}, and are pure rotations by an angle zero.
-The [[complex number|complex]] analogue of an orthogonal matrix is a [[unitary matrix]].
+The complex analogue of an orthogonal matrix is a unitary matrix.
 ===Main operations===

@@ Line 198: / Line 198: @@
 | Squeeze mapping<br>with ''r'' = 3/2
 | Scaling<br>by a factor of 3/2
-|<span id="rotation_matrix">Rotation<br>by {{pi}}/6 = 30°</span>
+|<span id="rotation_matrix">Rotation<br>by <math> \pi</math>/6 = 30°</span>
 |-
 | <math>\begin{bmatrix}