Elementary Linear Algebra

2012. 12. 16. 18:18

Coefficient matrix

The matrix A in this equation is called the coefficient matrix of the system.

Transpose

If A is any m x n matrix, then the transpose of A is defined to be the n x m matrix that results from interchanging the rows and columns of A.

Trace

If A is a square matrix, then the trace of A is defined to be the sum of the entries on the main diagonal of A.

Identity matrix

The n × n identity matrices, denoted by I_n, are square matrices with 1's on the main diagonal and 0's off the main diagonal.

Singular

If A is a square matrix, and if a matrix B of the same size can be found such that AB = BA = I, then A is said to be invertible and B is called an inverse of A.

If no such matrix B can be found, then A is said to be singular.

Unitary matrix

In mathematics, a complex square matrix U is unitary if

where I is the identity matrix and U^* is the conjugate transpose of U.

Elementary matrix

(http://en.wikipedia.org/wiki/Elementary_matrix)

An n x n matrix is called an elementary matrix if it can be obtained from the n x n identity matrix I_n by performing a single elementary row operation.

Row operation

Multiplying row i through by a nonzero constant c
Interchange rows i and j
Adding c times row i to row j

Row-echelon form

If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1.(leading 1)
If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.
If any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.

To be Reduced Row-echelon form,
Each column that contains a leading 1 has zeros everywhere else in that column.

Row-Echelon Form and Reduced Row-Echelon Form

Orthogonal Matrices

(http://en.wikipedia.org/wiki/Orthogonal_matrix)

A square matrix is orthogonal if its transpose is equal to its inverse.

Diagonal matrices

(http://en.wikipedia.org/wiki/Diagonal_matrix)

A square matrix in which all the entries off the main diagonal are zero.

Triangular matrices

A square matrix in which all the entries above the main diagonal are zero is called lower triangular, and a square matrix in which all the entries below the main diagonal are zero is called upper triangular.

3 × 3 upper triangular matrix and 3 × 3 lower triangular matrix

Symmetric matrices

A square matrix A is called symmetric if

Minors and Cofactors

If A is a square matrix, then the minor of entry a_ij is denoted by M_ij and is defined to be the determinant of the submatrix that remains after the ith row and jth column are deleted from A.

The number (-1)^i+jM_ij is denoted by C_ij and is called the cofactor of entry a_ij.