Returns the eigenvalues of A. The individual components of the factorization F can be accessed via getproperty: F further supports the following functions: lu! Note how the ordering of v and w differs on the left and right of these expressions (due to column-major storage). If order = B, eigvalues are ordered within a block. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. ipiv contains pivoting information about the factorization. (5.1) Lemma Let Abe an n×nmatrix containing a column of zeroes. If range = A, all the eigenvalues are found. If jobu = A, all the columns of U are computed. A is assumed to be symmetric. tau stores the elementary reflectors. If the underlying BLAS is using multiple threads, higher flop rates are realized. The permute, scale, and sortby keywords are the same as for eigen!. This is useful when optimizing critical code in order to avoid the overhead of repeated allocations. The LQ decomposition is the QR decomposition of transpose(A), and it is useful in order to compute the minimum-norm solution lq(A) \ b to an underdetermined system of equations (A has more columns than rows, but has full row rank). Calculate the matrix-matrix product $AB$, overwriting A, and return the result. (The kth generalized eigenvector can be obtained from the slice F.vectors[:, k].). The input matrices A and B will not contain their eigenvalues after eigvals! Determinant of diagonal matrix, triangular matrix (upper triangular or lower triangular matrix) is product of element of the principle diagonal. Compute the matrix sine and cosine of a square matrix A. Compute the matrix tangent of a square matrix A. Matrix division using a polyalgorithm. Here is why: expand with respect to that row. The solution is returned in B. Solves the linear equation A * X = B where A is a square matrix using the LU factorization of A. The result is stored in C by overwriting it. Return the largest eigenvalue of A. (Theorem 4.) The possibilities are: Dot product of two vectors consisting of n elements of array X with stride incx and n elements of array Y with stride incy. If uplo = U, the upper half of A is stored. A is the LU factorization from getrf!, with ipiv the pivoting information. If pivoting is chosen (default) the element type should also support abs and <. For the theory and logarithmic formulas used to compute this function, see [AH16_1]. x ⋅ y (where ⋅ can be typed by tab-completing \cdot in the REPL) is a synonym for dot(x, y). If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. select determines which eigenvalues are in the cluster. U, S, V and Vt can be obtained from the factorization F with F.U, F.S, F.V and F.Vt, such that A = U * Diagonal(S) * Vt. A is overwritten by its Cholesky decomposition. If you cannot find a row which makes the pivot non-zero, then the value of the determinant is zero, and further steps are not to be carried out. Return the solution to A*x = b or one of the other two variants determined by tA and ul. Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. When passed, jpvt must have length greater than or equal to n if A is an (m x n) matrix and tau must have length greater than or equal to the smallest dimension of A. Compute the RQ factorization of A, A = RQ. The vector v is destroyed during the computation. For instance: sB has been tagged as a matrix that's (real) symmetric, so for later operations we might perform on it, such as eigenfactorization or computing matrix-vector products, efficiencies can be found by only referencing half of it. For indefinite matrices, the LDLt factorization does not use pivoting during the numerical factorization and therefore the procedure can fail even for invertible matrices. Support for raising Irrational numbers (like ℯ) to a matrix was added in Julia 1.1. on A. Note that adjoint is applied recursively to elements. (c) 2 6 6 4 1 1 5 1 2 1 7 1 3 2 12 2 2 1 9 1 3 7 7 5. Matrix power, equivalent to $\exp(p\log(A))$. Iterating the decomposition produces the components U, S, and V. Matrix factorization type of the generalized singular value decomposition (SVD) of two matrices A and B, such that A = F.U*F.D1*F.R0*F.Q' and B = F.V*F.D2*F.R0*F.Q'. Another (typically slower but more accurate) option is alg = QRIteration(). A is assumed to be Hermitian. is the same as bunchkaufman, but saves space by overwriting the input A, instead of creating a copy. If sense = N, no reciprocal condition numbers are computed. Skeel condition number $\kappa_S$ of the matrix M, optionally with respect to the vector x, as computed using the operator p-norm. Returns T, Q, reordered eigenvalues in w, the condition number of the cluster of eigenvalues s, and the condition number of the invariant subspace sep. Reorders the vectors of a generalized Schur decomposition. If permuting was turned on, A[i,j] = 0 if j > i and 1 < j < ilo or j > ihi. The multi-cosine/sine matrices C and S provide a multi-measure of how much A vs how much B, and U and V provide directions in which these are measured. to divide scalar from left. Set the number of threads the BLAS library should use. Matrix factorization type of the Bunch-Kaufman factorization of a symmetric or Hermitian matrix A as P'UDU'P or P'LDL'P, depending on whether the upper (the default) or the lower triangle is stored in A. Compute the pivoted QR factorization of A, AP = QR using BLAS level 3. Solves the Sylvester matrix equation A * X +/- X * B = scale*C where A and B are both quasi-upper triangular. When constructed using qr, the block size is given by $n_b = \min(m, n, 36)$. The (quasi) triangular Schur factor can be obtained from the Schur object F with either F.Schur or F.T and the orthogonal/unitary Schur vectors can be obtained with F.vectors or F.Z such that A = F.vectors * F.Schur * F.vectors'. Overwrites B with the solution X and returns it. See the answer. If jobq = Q, the orthogonal/unitary matrix Q is computed. transpose(U) and transpose(L). Ferr is the forward error and Berr is the backward error, each component-wise. If uplo = L, the lower half is stored. If range = I, the eigenvalues with indices between il and iu are found. Returns X and the residual sum-of-squares. In a determinant each element in any row (or column) consists of the sum of two terms, then the determinant can be expressed as sum of two determinants of same order. If F::GeneralizedEigen is the factorization object, the eigenvalues can be obtained via F.values and the eigenvectors as the columns of the matrix F.vectors. For real vectors v and w, the Kronecker product is related to the outer product by kron(v,w) == vec(w * transpose(v)) or w * transpose(v) == reshape(kron(v,w), (length(w), length(v))). $$A, B) Matrix division using a polyalgorithm. A is overwritten by its Cholesky decomposition. if A == adjoint(A)). If uplo = U, the upper triangles of A and B are used. C is overwritten. = \prod_{j=1}^{b} (I - V_j T_j V_j^T)\], $\|A\|_p = \left( \sum_{i=1}^n | a_i | ^p \right)^{1/p}$, $\|A\|_1 = \max_{1 ≤ j ≤ n} \sum_{i=1}^m | a_{ij} |$, $\|A\|_\infty = \max_{1 ≤ i ≤ m} \sum _{j=1}^n | a_{ij} |$, \[\kappa_S(M, p) = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \right\Vert_p \\ If sense = E,B, the right and left eigenvectors must be computed. dA determines if the diagonal values are read or are assumed to be all ones. Returns the lower triangle of M starting from the kth superdiagonal. Question: Question 9 [10 Points] Compute The Determinant Of The Matrix A, Below, By Using Row Operations To Transform A To An Upper-triangular Matrix B. When A is not full rank, factorization with (column) pivoting is required to obtain a minimum norm solution. ipiv contains pivoting information about the factorization. If diag = N, A has non-unit diagonal elements. Transforms the upper trapezoidal matrix A to upper triangular form in-place. Compute the determinants of each of the following matrices: \(\begin{bmatrix} 2 & 3 \\ 0 & 2\end{bmatrix}$$ In fact, it is very easy to calculate the determinant of upper triangular matrix. If uplo = U, A is upper triangular. In most cases, if A is a subtype S of AbstractMatrix{T} with an element type T supporting +, -, * and /, the return type is LU{T,S{T}}. If range = I, the eigenvalues with indices between il and iu are found. jpvt is an integer vector of length n corresponding to the permutation $P$. Only the ul triangle of A is used. Equivalent to (log(abs(det(M))), sign(det(M))), but may provide increased accuracy and/or speed. The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., lu! To see the UniformScaling operator in action: If you need to solve many systems of the form (A+μI)x = b for the same A and different μ, it might be beneficial to first compute the Hessenberg factorization F of A via the hessenberg function. alpha is a scalar. Modifies the matrix/vector B in place with the solution. Returns the updated B. Return the upper triangle of M starting from the kth superdiagonal, overwriting M in the process. The individual components of the factorization F::LU can be accessed via getproperty: Iterating the factorization produces the components F.L, F.U, and F.p. Return Y. Overwrite X with a*X for the first n elements of array X with stride incx. Fact 6. tau must have length greater than or equal to the smallest dimension of A. Compute the RQ factorization of A, A = RQ. ), and performance-critical situations requiring rdiv! Same as eigvals, but saves space by overwriting the input A (and B), instead of creating copies. Solves A * X = B for positive-definite tridiagonal A. This is the return type of svd(_, _), the corresponding matrix factorization function. The second argument p is not necessarily a part of the interface for norm, i.e. If fact = F, equed may be N, meaning A has not been equilibrated; R, meaning A was multiplied by Diagonal(R) from the left; C, meaning A was multiplied by Diagonal(C) from the right; or B, meaning A was multiplied by Diagonal(R) from the left and Diagonal(C) from the right. C is overwritten. diagm constructs a full matrix; if you want storage-efficient versions with fast arithmetic, see Diagonal, Bidiagonal Tridiagonal and SymTridiagonal. For Adjoint/Transpose-wrapped vectors, return the operator $q$-norm of A, which is equivalent to the p-norm with value p = q/(q-1). Compute the operator norm (or matrix norm) induced by the vector p-norm, where valid values of p are 1, 2, or Inf. Only the ul triangle of A is used. Computes the Bunch-Kaufman factorization of a Hermitian matrix A. Those functions that overwrite one of the input arrays have names ending in '!'. Since the p-norm is computed using the norms of the entries of A, the p-norm of a vector of vectors is not compatible with the interpretation of it as a block vector in general if p != 2. p can assume any numeric value (even though not all values produce a mathematically valid vector norm). Construct a symmetric tridiagonal matrix from the diagonal and first superdiagonal of the symmetric matrix A. Construct a tridiagonal matrix from the first subdiagonal, diagonal, and first superdiagonal, respectively. alpha is a scalar. B is overwritten with the solution X and returned. is the same as qr when A is a subtype of StridedMatrix, but saves space by overwriting the input A, instead of creating a copy. Rank-2k update of the Hermitian matrix C as alpha*A*B' + alpha*B*A' + beta*C or alpha*A'*B + alpha*B'*A + beta*C according to trans. A is overwritten by its inverse. Return op(A)*b, where op is determined by tA. Lazy adjoint (conjugate transposition). The atol and rtol keyword arguments requires at least Julia 1.1. gels! See the picture below. Computes matrix N such that M * N = I, where I is the identity matrix. For A+I and A-I this means that A must be square. If $A$ is an m×n matrix, then. Solve the equation AB * X = B. trans determines the orientation of AB. A QR matrix factorization stored in a compact blocked format, typically obtained from qr. Update vector y as alpha*A*x + beta*y or alpha*A'*x + beta*y according to trans. A is overwritten by Q. If jobvt = O, A is overwritten with the rows of (thin) V'. A is assumed to be symmetric. Returns the solution X; equed, which is an output if fact is not N, and describes the equilibration that was performed; R, the row equilibration diagonal; C, the column equilibration diagonal; B, which may be overwritten with its equilibrated form Diagonal(R)*B (if trans = N and equed = R,B) or Diagonal(C)*B (if trans = T,C and equed = C,B); rcond, the reciprocal condition number of A after equilbrating; ferr, the forward error bound for each solution vector in X; berr, the forward error bound for each solution vector in X; and work, the reciprocal pivot growth factor. Return the singular values of A in descending order. Only the uplo triangle of C is used. 2.1.7 Upper triangular matrices Theorem 2.2. Use , , and keys on keyboard to move between field in calculator. Finds the singular value decomposition of A, A = U * S * V'. The difference in norm between a vector space and its dual arises to preserve the relationship between duality and the dot product, and the result is consistent with the operator p-norm of a 1 × n matrix. Entries of A below the first subdiagonal are ignored. Update C as alpha*A*B + beta*C or alpha*B*A + beta*C according to side. Matrix factorization type of the Cholesky factorization of a dense symmetric/Hermitian positive definite matrix A. Iterating the decomposition produces the components F.values and F.vectors. Returns C. Returns the uplo triangle of alpha*A*transpose(B) + alpha*B*transpose(A) or alpha*transpose(A)*B + alpha*transpose(B)*A, according to trans. R2 If one row is multiplied by ﬁ, then the determinant is multiplied by ﬁ. Explicitly finds the matrix Q of a QR factorization after calling geqrf! The result is of type SymTridiagonal and provides efficient specialized eigensolvers, but may be converted into a regular matrix with convert(Array, _) (or Array(_) for short). Otherwise, the inverse cosine is determined by using log and sqrt. Same as ldlt, but saves space by overwriting the input S, instead of creating a copy. abstol can be set as a tolerance for convergence. If sense = V, reciprocal condition numbers are computed for the right eigenvectors only. Finds the solution to A * X = B for Hermitian matrix A. They coincide at p = q = 2. The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals. matrix decompositions. $\left\vert M \right\vert$ denotes the matrix of (entry wise) absolute values of $M$; $\left\vert M \right\vert_{ij} = \left\vert M_{ij} \right\vert$. This is the return type of schur(_), the corresponding matrix factorization function. Prior to Julia 1.1, NaN and ±Inf entries in B were treated inconsistently. Use lmul! If jobvl = N, the left eigenvectors aren't computed. alpha and beta are scalars. A QR matrix factorization with column pivoting in a packed format, typically obtained from qr. In particular, this also applies to multiplication involving non-finite numbers such as NaN and ±Inf. If jobu = U, the orthogonal/unitary matrix U is computed. For symmetric or Hermitian A, an eigendecomposition (eigen) is used, otherwise the scaling and squaring algorithm (see [H05]) is chosen.