2012 MatrixComputations3rdEd

• (Golub & Van Loan, 2012) ⇒ Gene H. Golub, and Charles F. Van Loan. (2012). “Matrix Computations (4th Ed.).” Johns Hopkins University Press. ISBN:1421408597

Subject Headings: Matrix Computation, Cholesky factorization, Matrix Anal, QR Algorithm, Singular Value Decomposition, tridiagonal matrix.

Notes

• Google Books Book Page: http://books.google.com/books?id=X5YfsuCWpxMC
• Book Site: http://www.cs.cornell.edu/cv/GVL4/golubandvanloan.htm
• http://www.cs.cornell.edu/cv/GVL4/M-Files/M-Home.htm

Cited By

• http://scholar.google.com/scholar?q=%222012%22+Matrix+Computations

Quotes

Chapter 1 - Matrix Multiplication p.1

1.1 Basic Algorithms and Notation p.2

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Matrix computations are built upon a hierarchy of linear algebraic operations. Dot products involve the scalar operations of addition and multiplication. Matrix-vector multiplication is made up of dot products. Matrix-matrix multiplication amounts to a collection of matrix-vector products. All of these operations can be described in algorithmic form or in the language of linear algebra. One of our goals is to show how these two styles of expression complement each other. Along the way we pick up notation and acquaint the reader with the kind of thinking that underpins the matrix computation area. The discussion revolves around the matrix multiplication problem, a computation that can be organized in several ways.

1.1.1 Matrix Notation

Let designate the set of real numbers. We denote the vector space of all m-by-n real matrices by m ×n:

[math]\displaystyle{ }[/math]

If a capital letter is used to denote a matrix (e.g., A, B, Δ), then the corresponding lower case letter with subscript ij refers to the (i, j) entry (e.g., aij, bij, δij). Sometimes we designate the elements of a matrix with the notation [A] ij or A( i, j).

1.1.2 Matrix Operations

Basic matrix operations include transposition (m × n → n ×m), C = AT ⇒ cij = aij,

[math]\displaystyle{ C = A + B ⇒ cij = aij + bij, }[/math]

addition ([math]\displaystyle{ \R^{m × n} × \mathbb{R}^{m × n} → \mathbb{R}^{m × n} }[/math]),

[math]\displaystyle{ C = αA ⇒ cij = αaij, }[/math]

scalar-matrix multiplication ([math]\displaystyle{ \R × \mathbb{R}^{m × n} → \mathbb{R}^{m × n} }[/math]), and matrix-matrix multiplication ([math]\displaystyle{ \R^{m × p} × \mathbb{R}^{p × n} → \mathbb{R}^{m ×n} }[/math]),

[math]\displaystyle{ C = A B \Rightarrow c_{ij} = \Sigma^p_{k=1} }[/math]

Pointwise matrix operations are occasionally useful, especially pointwise multiplication ([math]\displaystyle{ \R^{m × n} × \mathbb{R}^{m × n} → \mathbb{R}^{m × n} }[/math]),

[math]\displaystyle{ C = A \ .^* B \Rightarrow c_{ij} = a_{ij} b_{ij} . }[/math]

and pointwise division ([math]\displaystyle{ \R^{m × n} × \mathbb{R}^{m × n} → \mathbb{R}^{m × n} }[/math]),

[math]\displaystyle{ C = A \ ./ B ⇒ c_{ij} = a_{ij} / b_{ij}. }[/math]

Of course, for pointwise division to make sense, the “denominator matrix” must have nonzero entries.

1.1.3 Vector Notation

Let [math]\displaystyle{ \R^n }[/math] denote the vector space of real n-vectors:

[math]\displaystyle{ x \in \mathbb{R}^n \Leftrightarrow \ \ x = \begin{bmatrix} x_{1} \\ \vdots \\ x_n \end{bmatrix} \ \ x_i \in \R. }[/math]

We refer to [math]\displaystyle{ x_i }[/math] as the ith component of [math]\displaystyle{ x }[/math]. Depending upon context, the alternative notations [math]\displaystyle{ [x]_i }[/math] and [math]\displaystyle{ x(i) }[/math] are sometimes used. Notice that we are identifying [math]\displaystyle{ \R^n }[/math] with [math]\displaystyle{ \R^{n × 1} }[/math] and so the members of [math]\displaystyle{ n }[/math] are column vectors. On the other hand, the elements of [math]\displaystyle{ n × 1 }[/math] are row vectors: [math]\displaystyle{ x \in \mathbb{R}^{n × 1} \Leftrightarrow x = [x_1,...,x_n] }[/math]. If [math]\displaystyle{ x }[/math] is a column vector, then [math]\displaystyle{ y = x^T }[/math] a row vector.

1.1.4 Vector Operations

Assume that a , x n, and y n. Basic vector operations include scalar-vector multiplication, z = ax ⇒ zi = axi, vector addition, z = x + y ⇒ zi = x_i + yi, and the inner product (or dot product), A particularly important operation, which we write in update form, is the saxpy: y = ax + y ⇒ yi = axi + yi Here, the symbol “[math]\displaystyle{ = }[/math]” is used to denote assignment, not mathematical equality. The vector y is being updated. The name “saxpy” is used in LAPACK , a software package that implements many of the algorithms in this book. “Saxpy” is a mnemonic for “scalar a x plus y.” See LAPACK.

[[Pointwise vector operation]s are also useful, including vector multiplication, [math]\displaystyle{ z = x \ .* y ⇒ z_i = x_iy_i }[/math], and vector division, </math>z = x \./ y ⇒ z_i = x_i / y_i</math>.

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References

• http://www.cs.cornell.edu/cv/GVL4/MasterSelect.pdf;

	Author	volume	Date Value	title	type	journal	titleUrl	doi	note	year
2012 MatrixComputations3rdEd	Gene H. Golub Charles F. Van Loan			Matrix Computations (3rd Ed.)						2012