Chemical decomposition, analysis or breakdown is the separation of a chemical compound into elements or simpler compounds. It is sometimes defined as the exact opposite of a chemical synthesis. Chemical decomposition is often an undesired chemical reaction. The stability that a chemical compound ordinarily has is eventually limited when exposed to extreme environmental conditions like heat, radiation, humidity or the acidity of a solvent. The details of decomposition processes are generally not well defined, as a molecule may break up into a host of smaller fragments. Chemical decomposition is exploited in several analytical techniques, notably mass spectrometry, traditional gravimetric analysis, and thermogravimetric analysis.

A broader definition of the term decomposition also includes the breakdown of one phase into two or more phases.

There are broadly three types of decomposition reactions: thermal, electrolytic and catalytic.

Reaction formula

The generalized reaction for chemical decomposition is:

AB → A + B

with a specific example being the electrolysis of water to gaseous hydrogen and oxygen:

2H₂O(I) → 2H₂ + O₂

Additional examples

An example of spontaneous decomposition is that of hydrogen peroxide, which will slowly decompose into water and oxygen:

2H₂O₂→ 2H₂O + O₂

Carbonates will decompose when heated, a notable exception being that of carbonic acid, H₂CO₃. Carbonic acid, the "fizz" in sodas, pop cans and other carbonated beverages, will decompose over time (spontaneously) into carbon dioxide and water

H₂CO₃→ H₂O + CO₂

Other carbonates will decompose when heated producing the corresponding metaloxide and carbon dioxide. In the following equation M represents a metal:

MCO₃→ MO + CO₂

A specific example of this involving calcium carbonate:

CaCO₃→ CaO + CO₂

Metal chlorates also decompose when heated. A metal chloride and oxygen gas are the products.

2MClO₃→ 2MCl + 3O₂

A common decomposition of a chlorate to evolve oxygen utilizes potassium chlorate as follows:

2KClO₃→ 2KCl + 3O₂

Many metal carbonates decompose to form metal oxides and carbon dioxide when heated.

Chemical decomposition, analysis or breakdown is the separation of a chemical compound into elements or simpler compounds. It is sometimes defined as the exact opposite of a chemical synthesis. Chemical decomposition is often an undesired chemical reaction. The stability that a chemical

In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.

Example

In numerical analysis, different decompositions are used to implement efficient matrix algorithms.

For instance, when solving a system of linear equations Ax=b, the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrixL and an upper triangular matrixU. The systems L(Ux)=b and Ux=L^{-1}b require fewer additions and multiplications to solve, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly the QR decomposition expresses A as QR with Q a unitary matrix and R an upper triangular matrix. The system Q(Rx) = b is solved by Rx = Q^Tb = c, and the system Rx = c is solved by "back substitution". The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable.

Decompositions related to solving systems of linear equations

• Applicable to: square matrixA
• Decomposition: A=LU, where L is lower triangular and U is upper triangular
• Related: the LDU decomposition is A=LDU, where L is lower triangular with ones on the diagonal, U is upper triangular with ones on the diagonal, and D is a diagonal matrix.
• Related: the LUP decomposition is A=LUP, where L is lower triangular, U is upper triangular, and P is a permutation matrix.
• Existence: An LUP decomposition exists for any square matrix A. When P is an identity matrix, the LUP decomposition reduces to the LU decomposition. If the LU decomposition exists, the LDU decomposition does too.
• Comments: The LUP and LU decompositions are useful in solving an n-by-n system of linear equations Ax=b. These decompositions summarize the process of Gaussian elimination in matrix form. Matrix P represents any row interchanges carried out in the process of Gaussian elimination. If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P=I, so an LU decomposition exists.

LU Reduction

Block LU decomposition

Rank factorization

• Applicable to: square, symmetric, positive definite matrix A
• Decomposition: A=U^TU, where U is upper triangular with positive diagonal entries
• Comment: the Cholesky decomposition is a special case of the symmetric LU decomposition, with L=U^T.
• Comment: the Cholesky decomposition is unique
• Comment: the Cholesky decomposition is also applicable for complex hermitian positive definite matrices
• Comment: An alternative is the LDL decomposition which can avoid extracting square roots.

• Applicable to: m-by-n matrix A
• Decomposition: A=QR where Q is an orthogonal matrix of size m-by-m, and R is an upper triangular matrix of size m-by-n
• Comment: The QR decomposition provides an alternative way of solving the system of equations Ax=b without inverting the matrix A. The fact that Q is orthogonal means that Q^TQ=I, so that Ax=b is equivalent to Rx=Q^Tb, which is easier to solve since R is triangular.

Singular value decomposition

• Applicable to: m-by-n matrix A.
• Decomposition: A=UDV^H, where D is a nonnegative diagonal matrix, and U and V are unitary matrices, and V^H denotes the conjugate transpose of V (or simply the transpose, if V contains real numbers