Matrix

A matrix represents a linear map after bases have been chosen for its domain and codomain.

A rectangular array whose entries are nondecreasing in each coordinate. Every subarray obtained by restricting to a row interval and a column interval has the same property. Comparing a target with the top-right entry discards a whole row or column, giving staircase matrix search in time linear in the sum of the dimensions.

A real symmetric matrix whose entries are the moments $H_{ij}={\int }_0^1{x}^{i-1}{x}^{j-1}dx$. Thus it is the Gram matrix of the monomials in the real L2 inner product on $[0,1]$. For a nonzero real coefficient vector $v$, $v^{T}Hv={\int }_0^1({\sum }_i{v}_i{x}^{i-1}{)}^{2}dx>0$, so every finite Hilbert matrix is positive-definite. The moment description connects these matrices with polynomial approximation and with the shifted Hilbert quadratic-form bound.

For real $a_r$, put $h(z)={\sum }_{r=1}^n{a}_r{z}^r$. The quadratic form equals ${\int }_0^{1}h(x{)}^{2}dx$, proving its nonnegativity. Apply the centred logarithmic keyhole identity bound to the entire function $h^2$. Since $|h(e^{i\theta }{)}^2|=|h(e^{i\theta }){|}^2$, orthogonality of the finite Fourier sum gives ${\int }_0^{2\pi }|h(e^{i\theta }){|}^{2}d\theta =2\pi {\sum }_r{a}_r^2$. Multiplication by the contour estimate's factor $1/2$ proves the upper bound. The matrix $(1/(r+s+1))$ is the principal submatrix of a Hilbert matrix obtained by deleting its first row and column.

The square matrix obtained by retaining the same index set $I$ of rows and columns of a square matrix $A$. Its determinant is a principal minor. A leading principal submatrix uses the first $k$ indices.

Evaluate a scalar polynomial on a square matrix by replacing its variable with that matrix and its constant term with a multiple of the identity matrix. Powers of the same matrix commute, so $p(A)q(A)=(pq)(A)$. For an eigenvector $Av=\lambda v$, $p(A)v=p(\lambda )v$.

A matrix representing observations and their measured variables. A common convention uses observations as rows and variables as columns; stating this orientation avoids transposition ambiguities. Column centering subtracts each variable's sample mean, and $X_c^T{X}_c/(n-1)$ is the sample covariance matrix of centered data when $n>1$.

A real matrix is a matrix whose entries are real numbers. The vector space of $m\times n$ real matrices is denoted $M_{m,n}(\mathbb{R})$.

A nonsymmetric matrix is a square matrix unequal to its transpose. Its right eigenvectors need not be orthogonal, and it need not be a diagonalizable matrix. For instance, $A=\left(\begin{array}{cc}1& 1\\ 0& 2\end{array}\right)$ has eigenvectors $(1,0{)}^T$ and $(1,1{)}^T$ for its distinct eigenvalues. A starting vector $(1,0{)}^T$ has nonzero inner product with $(1,1{)}^T$ but zero coefficient in that direction in the eigenbasis. This is why power iteration and inverse iteration require a nonzero spectral component rather than an ordinary nonorthogonality test in the general case.

Matrices of the same shape are added entry by entry. For fixed dimensions over a field, this operation and scalar multiplication make the matrices a vector space. A matrix product has a different dimension-matching rule: its inner dimensions must agree, whereas addition requires the entire shapes to agree.

A matrix whose entries are complex numbers. For an endomorphism of ${\mathbb{C}}^n$, Hermitian matrices and unitary matrices are distinguished by their behavior under the conjugate transpose.

An integer matrix is a matrix whose entries are integers. Sums and products of compatible integer matrices remain integer matrices, but a nonsingular square integer matrix can have a nonintegral inverse matrix. The unimodular matrix condition characterizes exactly when that inverse matrix also has integer entries.

A Cauchy matrix has entries $1/(x_i+y_j)$ with nonzero denominators. If $x_i=y_i>0$ are distinct, it is positive definite: $C_{ij}={\int }_0^{\infty }{e}^{-x_{i}s}{e}^{-x_{j}s}ds$ is the Gram matrix of linearly independent exponential functions. This form occurs in finite-rank inverse scattering transform reconstructions.

A matrix norm is a norm on a vector space of matrices; common choices include induced operator norms and the Frobenius norm. A submultiplicative matrix norm additionally satisfies $\Vert AB\Vert \le \Vert A\Vert \Vert B\Vert$. The matrix 2-norm is the norm induced by Euclidean norms and equals the largest singular value.

A vector norm on a finite-dimensional real or complex vector space induces the displayed matrix norm on its linear operators. It is the corresponding operator norm, and the supremum can equivalently be taken over the unit sphere. Applying the length bound twice proves submultiplicativity: $\Vert BCx\Vert \le \Vert B\Vert \Vert C\Vert \Vert x\Vert$, hence $\Vert BC\Vert \le \Vert B\Vert \Vert C\Vert$.

For each row, the weighted Cauchy-Schwarz inequality gives $|(Bx{)}_i{|}^2\le ({\sum }_j|b_{ij}|)({\sum }_j|b_{ij}||x_j{|}^2)$. Summing over rows bounds the result by the maximum absolute row sum times the maximum absolute column sum times $\Vert x{\Vert }_2^2$. Taking square roots and the supremum proves the displayed induced matrix norm inequality. It gives a Euclidean estimate directly from entries without computing singular values.

For a real symmetric positive-definite matrix $A$, the energy norm is $\Vert x{\Vert }_A=(x^{T}Ax{)}^{1/2}=\Vert A^{1/2}x{\Vert }_2$. In the defining supremum of the induced matrix norm, substitute $y=A^{1/2}x$. The matrix square root is an invertible matrix, so this ranges over every nonzero vector and proves the displayed identity. This matrix similarity is useful because contraction in the energy norm becomes an ordinary Euclidean operator norm estimate after a change of coordinates.

A matrix pencil is an affine one-parameter family of matrices. For square matrices of size $m$, its determinant is a polynomial of degree at most $m$. If $B$ is invertible, its leading coefficient is $\det B\ne 0$, so at most $m$ distinct parameters give a singular matrix. This translates linear dependence in a family of bases into polynomial root counting.

A real rectangular matrix is totally nonnegative if every square minor, with row and column indices in increasing order, is nonnegative. A nonnegative bidiagonal matrix is totally nonnegative, and products of compatible totally nonnegative matrices have the same property by the Cauchy–Binet formula. This controls signs of inverse entries and underlies stability and shape-preserving properties of B-spline collocation. Total nonnegativity permits zero minors; strict total positivity requires every such minor to be positive.

If an invertible square totally nonnegative matrix $A$ has size $n$, then $\det A>0$ andIndeed the adjugate identity gives $(A^{-1}{)}_{ij}=(-1{)}^{i+j}\det A_{\hat{j},\hat{i}}/\det A$, and the deleted-row/deleted-column minor is nonnegative. Consequently an alternating vector $v_j=(-1{)}^j$ gives $(A^{-1}v{)}_i=(-1{)}^i{\sum }_j|(A^{-1}{)}_{ij}|$.

An elementary column operation interchanges two columns of a matrix, scales one by a nonzero scalar, or adds a multiple of another column. It is an elementary row operation on the transpose. Hence it has the same three effects on the determinant: changing sign, multiplying by the scale factor, or leaving the determinant unchanged.

The reduced column echelon form of a matrix $A$ is defined by taking its transpose, forming the reduced row echelon form, and transposing back:It is obtained from $A$ by elementary column operations. Its nonzero columns precede its zero columns, the topmost nonzero entry of each nonzero column is $1$, these leading entries move strictly downward as the column number increases, and each leading entry is the only nonzero entry in its row.

Every elementary column operation preserves the column space, because it and its inverse express each new column as a linear combination of the old columns. Thus the nonzero columns of the reduced form give a basis of the original column space. In contrast, elementary row operations preserve the row space. There is no general identity $\mathrm{rcef}(A)=\mathrm{rref}(A{)}^T$: the definition requires reducing $A^T$, not $A$.

For an invertible matrix of size $n$, both reduced forms are $I_n$. A singular example showing the distinction is

A Hadamard matrix is a real square matrix with entries $+1$ and $-1$ whose rows are pairwise orthogonal. Each row has squared norm $n$, so $H{H}^T=nI$ and $H/\sqrt{n}$ is an orthogonal matrix. In particular $H$ is invertible, with $H^{-1}=H^T/n$. The four-by-four exampleis symmetric and satisfies $H^2=4I$. It supplies an exact intertwiner in four-tile mixed-boundary transplantation between a disk and a nonorientable surface.

A matrix with the same number of rows and columns represents an endomorphism of a finite-dimensional vector space after choosing a basis. Square matrices support trace, determinant, powers and the matrix exponential.

For $A=P+iQ$ with real matrices $P,Q$, the same linear map viewed over the reals has block matrix $\left(\begin{array}{cc}P& -Q\\ Q& P\end{array}\right)$ in grouped real/imaginary coordinates. Interleaved coordinates merely permute the basis. Its complexification splits into the matrices $A$ and $\overline{A}$, giving the realification determinant identity.

The realification of a complex matrix has real determinant $|{\det }_{\mathbb{C}}A{|}^2$. Indeed the invertible complex coordinate change $(x,y)\mapsto (x+iy,x-iy)$ conjugates its complexified block matrix to $\mathrm{diag}(A,\overline{A})$. Taking determinants gives the identity, including singular matrices.

For a real matrix $M$, the entrywise L1 norm is ${\sum }_{r,s}|M_{rs}|$, the L1 norm of its vector of entries. It is the usual penalty in the all-entry convention for the Graphical Lasso.

Applying a scalar function separately to each matrix entry: $(f[X]{)}_{ij}=f(X_{ij})$. This differs from spectral matrix functional calculus. In particular, $\arcsin [Y]$ in Gaussian hyperplane rounding is entrywise, and its contribution to a matrix trace can be restricted to selected blocks by the support of the other matrix.

The matrix with entries $(P\circ Q{)}_{ij}=P_{ij}{Q}_{ij}$. It differs from ordinary matrix multiplication. The Schur product theorem states that it preserves positive semidefiniteness when both factors are positive semidefinite matrices.

For integer $k\ge 1$, the entrywise power has entries $X_{ij}^k$ and equals the repeated Hadamard product. The zero power is the all-ones matrix, even at zero entries, because it represents the constant function one. Every nonnegative integer Hadamard power of a positive semidefinite matrix is positive semidefinite by the Schur product theorem. It is not an ordinary matrix power.

The Hermite normal form is a canonical triangular form for an integer matrix under integer elementary row operations or, in the column convention, column operations. In the row convention for a nonsingular rank-two matrix, positive diagonal entries and reduction of the upper-right entry modulo the lower diagonal entry give the row Hermite normal form in rank two.

The row lattice of an integer matrix with $m$ rows is the set of their integer linear combinations. It is an Euclidean lattice in its real span, since it is contained in the discrete set ${\mathbb{Z}}^n$ and spans that real vector space. For a nonsingular two-by-two integer matrix, the row Hermite normal form in rank two gives a canonical basis and proves that its index in ${\mathbb{Z}}^2$ is the absolute value of its determinant.

For a finite-index subgroup $L\le {\mathbb{Z}}^2$, its projection to the first coordinate is $a\mathbb{Z}$ and its intersection with the second axis is $\{0\}\times d\mathbb{Z}$, with $a,d>0$. Choose $(a,b)\in L$, reducing $b$ modulo $d$. Together with $(0,d)$ it is a basis of this row lattice: subtracting a multiple of $(a,b)$ from any vector leaves a vector on the second axis. These parameters are unique, and reduction of the two coordinates shows $[{\mathbb{Z}}^2:L]=ad$. Consequently an integral matrix of positive determinant $n$ has a unique representative of this form under left multiplication by $S{L}_2(\mathbb{Z})$, with $ad=n$. This proves the lattice facts underlying determinant-n matrix representatives for Hecke operators.

A square real matrix is doubly stochastic if its entries are nonnegative and every row and column sums to $1$. Its positive-entry support satisfies the condition of the Hall marriage theorem: for a row subset $S$, $|S|\le |N(S)|$ by summing column capacities. Consequently its support contains a permutation matrix.

Let $A$ be a doubly stochastic matrix, let $q$ be a probability vector, and put $w=Aq$. In row $i$ with $w_i>0$, define $r_i(j)=a_{ij}{q}_j/w_i$ on the support where $a_{ij}>0$. The positive row entries ${\alpha }_i=(a_{ij})$ form a probability vector. Expanding relative entropy and using the column sums givesZero-output rows contribute zero because all their $a_{ij}{q}_j$ vanish. The row sums supply the normalized comparison laws, while the column sums preserve the input entropy contribution. Equality holds exactly when $q_j=w_i$ for every positive matrix entry $a_{ij}$, including the automatic zero values in zero-output rows.

This doubly stochastic matrix replaces two values $a,b$ by $ta+(1-t)b$ and $(1-t)a+tb$, preserving their sum. It fixes all other coordinates when embedded in a larger matrix. A finite product of such averaging matrices and permutation matrices is again doubly stochastic.

Every doubly stochastic matrix is a convex combination of permutation matrices. Equivalently, the extreme points of the doubly stochastic convex polytope are exactly the permutation matrices. The extreme-point assertion follows from an alternating-cycle perturbation of a doubly stochastic matrix; the convex-hull assertion follows because a bounded finite-dimensional polytope is the convex hull of its vertices. Permutation matrices are extreme since their zero entries force the same zeros in every convex decomposition.

For a doubly stochastic matrix, the Hall marriage theorem supplies a perfect matching in its positive-entry support. Subtract the smallest matched entry times the corresponding permutation matrix. The residual has equal row and column sums and smaller support unless it vanishes. Repeating gives a convex combination of permutation matrices. With $r$ initially positive entries, there are at most $r-n+1$ terms, since a positive-mass residual has at least $n$ positive entries. Elementary augmenting-path matching gives $O(n^5)$ arithmetic operations.

For a doubly stochastic matrix with a fractional entry, form the row-column bipartite graph of entries strictly between zero and one. Every incident vertex has degree at least two, so there is an even cycle. A matrix supported on that cycle with alternating entries $+1,-1$ has zero row and column sums. A sufficiently small positive multiple can be both added and subtracted while preserving entry bounds. The original matrix is then the midpoint of two distinct doubly stochastic matrices, so it is not an extreme point.

A rational matrix is a matrix whose entries are rational numbers. Multiplying by a common denominator produces an integer matrix and preserves its homogeneous solution set.

A real matrix is nonnegative when every entry is nonnegative. This entrywise condition differs from being a positive semidefinite matrix: the symmetric matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ is nonnegative but has eigenvalues $1$ and $-1$.

A square nonnegative matrix $A$ is irreducible if for each pair $(i,j)$ there is an integer $k\ge 0$ with $(A^k{)}_{ij}>0$. This says that every index can reach every other through positive entries. The Perron–Frobenius theorem then gives a positive leading eigenvector and an algebraically simple eigenvalue. Irreducibility alone permits other eigenvalues of the same modulus: the cyclic permutation matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ has eigenvalues $1,-1$.

A square nonnegative matrix $A$ is primitive if some positive integer power $A^k$ has strictly positive entries. It is therefore an irreducible nonnegative matrix. The Perron–Frobenius theorem gives a leading eigenvalue whose modulus is strictly larger than that of every other eigenvalue. Every strictly positive matrix is primitive; the two-cycle permutation matrix is irreducible but not primitive.

The strictly positive case states that a real square matrix $W$ with $W_{ij}>0$ has a positive simple eigenvalue $\lambda$ and positive left and right eigenvectors, with all other eigenvalues strictly smaller in modulus. The Brouwer fixed-point theorem applied to $v\mapsto Wv/(1^{T}Wv)$ on the nonnegative unit simplex gives $v>0$ and $Wv=\lambda v$. For another eigenvector $z$, maximize $|z_i|/v_i$; the triangle inequality gives $|\mu |\le \lambda$. Equality forces every modulus ratio and complex argument to agree because every entry is positive, so $z$ is proportional to $v$ and $\mu =\lambda$. Apply the same argument to $W^T$ for a positive left eigenvector; its positive pairing with $v$ rules out a generalized eigenvector at $\lambda$, establishing algebraic simplicity. For merely nonnegative matrices a nonnegative leading eigenvector exists. An irreducible nonnegative matrix has a positive leading eigenvector and simple Perron eigenvalue; a primitive nonnegative matrix has the strict modulus gap.

The all-ones matrix $J=\mathbf{e}{\mathbf{e}}^T$ has every entry equal to one, where $\mathbf{e}=(1,\dots ,1{)}^T$. On ${\mathbb{R}}^n$, the matrix $J/n$ is the orthogonal projection matrix onto $\mathrm{span}\{\mathbf{e}\}$.

For $u=(1,\dots ,1{)}^T$, $J_n=u{u}^T$ maps $v$ to $({\sum }_i{v}_i)u$. Therefore $u$ is an eigenvector with eigenvalue $n$, while the zero-sum vector subspace is its kernel. For $n\ge 2$ a basis of the kernel is $e_1-e_n,\dots ,e_{n-1}-e_n$. These vectors and $u$ form an eigenbasis, proving diagonalization and characteristic polynomial $t^{n-1}(t-n)$. Both algebraic multiplicity and geometric multiplicity are one at $n$ and $n-1$ at zero. At $n=1$ the sole eigenvalue is 1; the zero-sum space has dimension zero and contributes no additional root.

The row space of a matrix is the span of its rows, and its column space is the span of its columns. Both have dimension equal to the matrix rank, though they lie in different ambient coordinate spaces for a rectangular matrix.

The row space of a matrix is the linear span of its rows. It is the column space of the matrix transpose.

The column space of a matrix $A$ is the linear span of its columns. It is the image of a linear map represented by $A$.

If an $m\times n$ matrix has columns $c_1,\dots ,c_n$, then matrix multiplication gives $Ax=x_1{c}_1+\cdots +x_n{c}_n$. Thus its column space consists of exactly the attainable output vectors in ${\mathbb{F}}^m$, and $Ax=b$ has a solution exactly when $b$ belongs to this column space. Its dimension is the matrix rank; the kernel instead describes input vectors in ${\mathbb{F}}^n$ whose contributions cancel.^[ref]

A basis of a matrix's column space. Row reduction identifies which original columns to select: take the original columns at the pivot columns of the reduced matrix. Row operations preserve column dependence relations, but generally change the column vectors and their span. Choosing columns of the reduced matrix instead therefore need not give the original column space.

A binary matrix is a matrix all of whose entries belong to $\{0,1\}$.

A submatrix is obtained from a matrix by retaining selected rows and selected columns. A principal submatrix of a square matrix retains the same index set for its rows and columns.

A matrix minor is the determinant of a square submatrix obtained by selecting rows and columns of a matrix. Over a field, the matrix rank is less than $r$ exactly when every $r$-by-$r$ matrix minor vanishes. In a matrix of regular functions, these equations define a closed rank-defect locus, explaining the determinantal variety construction.

A principal minor is the determinant of a square submatrix selected using the same index set for rows and columns. For a three-by-three matrix, the coefficient of $\lambda$ in its monic characteristic polynomial is the sum of its three principal minors of order two.

A leading principal minor is a principal minor using the first $k$ rows and columns of a square matrix in a specified ordering. Sylvester's criterion uses these minors to characterize positive-definite matrices.

The entrywise maximum norm is the largest absolute value of a matrix entry. For compatible vectors $x$ and $y$,

A tridiagonal matrix has zero entries outside its main diagonal and the two adjacent diagonals. Linear systems with a nonsingular tridiagonal coefficient matrix can be solved in linear time by the Thomas algorithm.

A matrix decomposition expresses a matrix as a matrix product of factors with simpler structure. An LU decomposition uses triangular factors, a QR decomposition uses an orthogonal matrix and a triangular factor, and a singular value decomposition separates input directions, stretching factors and output directions. The factorization is an identity; the chosen structure determines what it helps compute.

For an original two-by-two example,The outer factors are a lower triangular matrix and an upper triangular matrix, each with unit diagonal as in a unit triangular matrix; the middle factor is a diagonal matrix. Their matrix product is $M$, and the final diagonal entry $5/2$ is the Schur complement of the first entry $2$. This is the scalar version of the factorization under block elimination.

To solve $Mz=(3,4{)}^T$, solve with the three factors in their displayed order from left to right: first the lower triangular factor, then the diagonal factor, then the upper triangular factor. The intermediate vectors are $(3,5/2{)}^T$ and $(3/2,1{)}^T$, giving the concise answer $z=(1,1{)}^T$. A new right-hand side uses the same factors. This is why Reusing LU factors for multiple right-hand sides is useful: factorization records work that otherwise would be repeated, while forward substitution and back substitution apply it cheaply.^[ref]

Different decompositions reveal different features. A Cholesky decomposition uses positive definiteness of a symmetric matrix, while a singular value decomposition also applies to rectangular matrices and exposes matrix rank. The existence and numerical behaviour of the factors depend on their own hypotheses; for example, an unpivoted LU decomposition can fail when a chosen pivot element is zero even though the matrix is an invertible matrix.

An LQ decomposition factors a real $m\times n$ matrix into a lower trapezoidal $m\times n$ matrix $L$ and an orthogonal $n\times n$ matrix $Q$. Transposing a QR decomposition of $A^T$ proves existence. Equivalently, right Householder transformations clear the part of each row beyond its diagonal while preserving earlier zeros. This factorization is useful for underdetermined systems, complementing the role of QR decomposition in overdetermined systems.

A block matrix partitions its rows and columns into rectangular submatrices called blocks, allowing a matrix calculation to be expressed in terms of those blocks.

A common symmetric saddle-point system couples primal unknowns with constraint multipliers through the displayed block matrix, with symmetric $H,C$. At $C=0$, full row rank of $A$ and a nonsingular reduced Hessian are exactly the nonsingularity conditions. Such matrices arise from equality-constrained quadratic models and mixed finite element methods. A regularized block $C=\mu I$ also arises when an augmented linear system replaces a badly scaled quadratic-penalty Newton system.

A block matrix is block diagonal if all off-diagonal blocks vanish. It represents independent actions on a direct sum of vector spaces. Its determinant is the product of the diagonal-block determinants.

A block tridiagonal matrix has nonzero blocks only on its main block diagonal and the two adjacent block diagonals. It is the block analogue of a tridiagonal matrix.

A square matrix is diagonally dominant when the absolute value of every diagonal entry is at least the sum of the absolute values of the other entries in its row.

A matrix is strictly diagonally dominant when every diagonal-dominance inequality is strict. A real symmetric strictly diagonally dominant matrix with positive diagonal is positive definite by the Gershgorin circle theorem.

If $\delta ={\min }_i(|a_{ii}|-{\sum }_{j\ne i}|a_{ij}|)>0$, then $A$ is invertible and its operator norm on ${\ell }^{\infty }$ satisfies the displayed bound. For $Ax=b$, choose $i$ maximizing $|x_i|$. The reverse triangle inequality gives $|b_i|\ge \delta \Vert x{\Vert }_{{\ell }^{\infty }}$. Taking $b=0$ proves injectivity and hence invertibility in finite dimensions; taking arbitrary $b$ proves the bound. The estimate is useful for converting strict diagonal dominance into quantitative matrix inverse stability.

The matrix transpose interchanges rows and columns: $(A^T{)}_{ij}=A_{ji}$. On a fixed matrix space, $A\mapsto A^T$ is the transposition map, a linear map that preserves the trace norm but is not completely positive in dimensions greater than one.

For real rectangular matrices, expanding $(A+H{)}^t(A+H)$ leaves the quadratic remainder $H^{t}H$, whose matrix norm is $O(\Vert H{\Vert }^2)$. Thus the displayed Fréchet derivative is a linear map in the increment $H$. At an orthogonal matrix, its kernel consists of increments $H=A\Omega$ with $\Omega$ skew-symmetric. This is the tangent-space condition obtained by differentiating $A^{t}A=I$.

For compatible finite matrices, the transpose reverses the factor order. Entrywise, $[(AB{)}^T{]}_{ij}={\sum }_k{A}_{jk}{B}_{ki}={\sum }_k(B^T{)}_{ik}(A^T{)}_{kj}$, proving the identity. The analogous formula for conjugate transpose is $(AB{)}^{†}=B^{†}{A}^{†}$. Commutativity of scalar entries permits the last rearrangement; the matrix products themselves are not assumed to commute.

Over a field of characteristic two, the linear map $T:M_n(F)\to M_n(F)$, $A\mapsto A+A^T$, has kernel of a linear map the symmetric matrices, of dimension $n(n+1)/2$. Its image of a linear map is the linear subspace of symmetric matrices with zero diagonal, of dimension $n(n-1)/2$. Indeed, $T(A)=0$ means $A=A^T$, and every symmetric zero-diagonal matrix is the image of the matrix obtained by retaining just its entries above the diagonal. Thus the two dimensions still add to $n^2$, as required by the rank-nullity theorem.

For a matrix over the real numbers or complex numbers, applying the transpose twice to $A=\lambda A^T$ gives $A={\lambda }^{2}A$. A nonzero matrix is therefore possible only for $\lambda =1$ or $\lambda =-1$. These cases are precisely the symmetric matrices and skew-symmetric matrices, respectively. For every other scalar, the only solution is the zero matrix.

For a positive-definite Hermitian matrix with spectral decomposition $A=U\mathrm{diag}({\lambda }_j)U^{\ast }$, the matrix logarithm is $\log A=U\mathrm{diag}(\log {\lambda }_j)U^{\ast }$. More generally it is a branch of the inverse of the matrix exponential when such a branch exists.

On complex symmetric matrices with positive-definite real part, the determinant has a nonzero analytic square root normalized positively on real positive matrices. It is continued within this accretive domain. For $A=G+iB$, put $C=G^{-1/2}B{G}^{-1/2}$ with real eigenvalues $b_j$; then $d(A)=\sqrt{\det G}{\prod }_j\sqrt{1+i{b}_j}$, using positive-real-part roots for each factor. The principal scalar square root of the product determinant need not equal this continued root: accumulated determinant arguments can cross the scalar branch cut. The principal matrix logarithm instead tracks all factors consistently.

For a positive-definite Hermitian operator $A={\sum }_j{t}_j|u_j⟩⟨u_j|$ and a unit vector $v$, the weights $|⟨v|u_j⟩{|}^2$ form a probability distribution. Scalar logarithmic concavity bounds their average of $\log t_j$ by the logarithm of their average of $t_j$. Positive semidefinite operators follow by a limiting convention. Using the eigenbasis of a density operator $\rho$ gives $S(\rho \Vert \sigma )\ge D(r\Vert q)$, where $r$ is the spectrum of $\rho$ and $q$ the diagonal of $\sigma$ in that basis. This yields nonnegativity of quantum relative entropy without commutativity.

An invertible complex Jordan block $\lambda I+N$, with $N^s=0$, has logarithm $\ell I+{\sum }_{k=1}^{s-1}(-1{)}^{k+1}(N/\lambda {)}^k/k$, where $e^{\ell }=\lambda$. Applying this construction blockwise and conjugating back proves surjectivity of the matrix exponential on the complex general linear group. There is no globally single-valued continuous choice of logarithm.

For positive-definite Hermitian matrices $A,B$, Klein's inequality giveswith equality exactly when $A=B$. To prove it, let $a_i,b_j$ be their eigenvalues and $u_i,v_j$ their orthonormal eigenvectors. The weights $w_{ij}=|⟨u_i,v_j⟩{|}^2$ have row and column sums one. The difference between the two sides isby the scalar logarithm inequality $\ln t\le t-1$. Equality requires $a_i=b_j$ whenever $w_{ij}>0$, implying $A=B$. Limits extend the result to positive semidefinite matrices with the appropriate support condition. Applying it to trace-one matrices proves nonnegativity of quantum relative entropy.

For a square matrix $A$ and a nonnegative integer $k$, the matrix power is the repeated productwith $A^0=I$.

A cube root of a matrix $A$ is a square matrix $B$ of the same size over the specified field satisfying $B^3=A$. Every complex invertible matrix has one: put it in Jordan normal form, choose a scalar cube root of each nonzero eigenvalue, and apply the truncated binomial cube root of the identity plus a nilpotent matrix on each block. The Nilpotent Jordan block $J_3(0)$ has none, even though its square is nonzero. If $B^3=J_3(0)$, then $B^9=0$, making $B$ a nilpotent matrix of size three; the Cayley-Hamilton theorem then gives $B^3=0$, a contradiction.

Compute a nonnegative integer power $A^n$ by repeated squaring. If $n=2k$, use $A^n=(A^k{)}^2$; if $n=2k+1$, use $A^n=A(A^k{)}^2$. This needs $O(\log n)$ matrix multiplications and permits fast evaluation of fixed-order linear recurrences via a companion matrix. It is distinct from computing the matrix exponential $e^A$.

If a diagonalizable matrix has distinct eigenvalues $r,s$, its spectral projections of a diagonalizable matrix are $(A-sI)/(r-s)$ and $(rI-A)/(r-s)$. Raising the diagonal form to a nonnegative integer power gives the displayed formula. Equivalently, interpolate the scalar function $t^n$ by a degree-one polynomial at $r,s$, then apply it to $A$. The diagonalizability assumption matters: knowledge of two eigenvalues alone does not exclude nontrivial Jordan blocks.

A matrix element $A_{ij}$ is the entry in row $i$ and column $j$ of a matrix $A$.

For a linear operator $T$ on an inner product space with an orthonormal basis $(e_i)$, its matrix elements are $T_{ij}=⟨e_i,T{e}_j⟩$, using the convention that the inner product is linear in its second argument. For a finite-dimensional space, the expansion $T{e}_j={\sum }_i{T}_{ij}{e}_i$ explains why these numbers represent the linear operator in that basis. In quantum mechanics, the expression $⟨\varphi |T|\psi ⟩$ is also called a matrix element between the two states; it need not be diagonal or an expectation value.

A matrix element of an interaction operator between hadron states. It contains strong interaction dynamics beyond the explicit CKM matrix factors of a weak interaction decay; its magnitude and strong phase need not be identical for two different channels.

A sparse matrix has few nonzero matrix elements in each row or column compared with its dimension. Algorithms normally require an efficient oracle that lists the positions and values of those entries.

A band matrix has nonzero entries confined to a band around its main diagonal. The nonnegative integers $p,q$ in the displayed condition are its lower and upper bandwidths. A tridiagonal matrix has $p=q=1$. Storing only the band reduces memory use; banded Gaussian elimination can exploit the same structure when the pivoting assumptions permit it.

Compressed sparse storage keeps a sparse matrix as numerical nonzero values, their row or column indices, and pointers to the start of each compressed column or row. It avoids allocating all $N^2$ entries. A sparse direct solver must additionally allocate possible fill-in predicted by symbolic factorization, because elimination can create entries absent from the input.

The bandwidth in this convention is the largest distance from the main diagonal of a nonzero matrix entry. A symmetric positive-definite matrix of bandwidth $b$ can be factored using $O(N{b}^2)$ operations and $O(Nb)$ storage; its Cholesky decomposition does not introduce entries outside that band. A variable permutation changes bandwidth, so it is a property of both the sparsity pattern and its ordering.

The matrix unit $E_{ij}$ has entry one in row $i$, column $j$, and zero in every other position. The matrix units form the standard basis of the vector space of matrices of a fixed size.

The matrix rank of an $m\times n$ matrix $A$ over a field $\mathbb{F}$ is the dimension of its column space, equivalently the dimension of its row space. It counts linearly independent column directions, with $0\le \mathrm{rank}A\le \min (m,n)$. Counting nonzero matrix entries gives a different quantity: an all-ones matrix can have arbitrarily many nonzero matrix entries and still have matrix rank one.^[ref]

Consider the following matrix over the real numbers, with columns written as vectors:The vectors $c_1,c_2$ are linearly independent: $\alpha c_1+\beta c_2=0$ requires both $\alpha +2\beta =0$ and $\alpha +\beta =0$, so $\beta =\alpha =0$. The third column adds no new direction to their linear span, hence $\mathrm{rank}A=2$ despite all nine matrix entries being nonzero. The second row likewise repeats twice the first row.

The column space is a plane through the origin:Every output has second coordinate twice its first coordinate. Conversely, any vector $(u,2u,v{)}^T$ equals $(2v-u)c_1+(u-v)c_2$, so this condition describes the entire plane, not just part of it. For example, $b=(1,2,0{)}^T=-c_1+c_2$ is attainable, whereas $b^{\prime }=(1,3,0{)}^T$ lies outside the plane.