Matrix decomposition

Key takeaways

Matrix decomposition belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Matrix decomposition to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Matrix decomposition from memory before moving on to harder problems.

Reference excerpt

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. 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 example, when solving a system of linear equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } , the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems L ( U x ) = b {\displaystyle L(U\mathbf {x} )=\mathbf {b} } and U x = L − 1 b {\displaystyle U\mathbf {x} =L^{-1}\mathbf {b} } require fewer additions and multiplications to solve, compared with the original system A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } , though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses A as QR with Q an orthogonal matrix and R an upper triangular matrix. The system Q(Rx) = b is solved by Rx = QTb = 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

LU decomposition

Traditionally applicable to: square matrix A, although rectangular matrices can be applicable. Decomposition: A = L U {\displaystyle A=LU} , where L is lower triangular and U is upper triangular. Related: the LDU decomposition is A = L D U {\displaystyle 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 P A = L U {\displaystyle PA=LU} , 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. Comments: The LUP and LU decompositions are useful in solving an n-by-n system of linear equations A x = b {\displaystyle A\mathbf {x} =\mathbf {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: m-by-n matrix A of rank r Decomposition: A = C F {\displaystyle A=CF} where C is an m-by-r full column rank matrix and F is an r-by-n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A, which one can apply to obtain all solutions of the linear system A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } .

Cholesky decomposition

Applicable to: square, hermitian, positive definite matrix A {\displaystyle A}

Decomposition: A = U ∗ U {\displaystyle A=U^{*}U} , where U {\displaystyle U} is upper triangular with real positive diagonal entries Comment: if the matrix A {\displaystyle A} is Hermitian and positive semi-definite, then it has a decomposition of the form A = U ∗ U {\displaystyle A=U^{*}U} if the diagonal entries of U {\displaystyle U} are allowed to be zero Uniqueness: for positive definite matrices Cholesky decomposition is unique. However, it is not unique in the positive semi-definite case. Comment: if A {\displaystyle A} is real and symmetric, U {\displaystyle U} has all real elements Comment: An alternative is the LDL decomposition, which can avoid extracting square roots.

QR decomposition

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Matrix decomposition

Start with the simplest possible case. Write down what Matrix decomposition claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Matrix decomposition before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Matrix decomposition ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Matrix decomposition

In research: Matrix decomposition appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Matrix decomposition in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Matrix decomposition is common in secondary-school and first-year university syllabi. It links to neighbouring topics Factorization, Matrix decompositions, Matrix theory, so understanding it makes those chapters shorter.
In everyday life: Look for Matrix decomposition outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Matrix decomposition in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Matrix decomposition means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Matrix decomposition out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Matrix decomposition in simple terms?

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

Why does Matrix decomposition matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Matrix decomposition?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Matrix decomposition.