proofthe matrix is both hermitian (by definitionand real (by assumptionand hence it is symmetric by theorem we can write qdqwhere is the diagonal matrix of (realeigenvalues of by theorem all eigenvalues are non-negativeand thus their square root is real-valued nowdefine dwhere is defined as the diagonal matrix whose elements are the square roots of the eigenvalues of diagonal thenbb dq qdqa the converse statement follows from the fact that xbbx kbxk for all matrix decompositions matrix decompositions are frequently used in linear algebra to simplify proofsavoid numerical instabilityand to speed up computations we mention three important matrix decompositions( )luqrand svd ( )lu decomposition every invertible matrix can be written as the product of three matricesa plupermutation matrix plu decomposition ( where is lower triangular matrixu an upper triangular matrixand permutation matrix permutation matrix is square matrix with single in each row and columnand zeros otherwise the matrix product pb simply permutes the rows of matrix andlikewisebp permutes its columns decomposition of the form ( is called plu decomposition as permutation matrix is orthogonalits transpose is equal to its inverseand so we can write ( as pa lu the decomposition is not uniqueand in many cases can be taken to be the identity matrixin which case we speak of the lu decomposition of aalso called the lr for left-right (triangulardecomposition plu decomposition of an invertible matrix can be obtained recursively as follows the first step is to swap the rows of such that the element in the first column and first row of the pivoted matrix is as large as possible in absolute value write the resulting matrix as where is the permutation matrix that swaps the first and -th rowwhere is the row that contains the largest element in the first column nextadd the matrix - [ > / to the last rows of to obtain the matrix > > = > / in effectwe add some multiple of the first row to each of the remaining rows in order to obtain zeros in the first columnexcept for the first element we now apply the same procedure to as we did to and then to subsequent smaller matrices an-