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APPENDIX A. A primer on linear matrix algebra

Note: this is a simplified presentation for finite-dimensional real vector spaces. For more general results and rigorous mathematical definitions, refer to mathematical textbooks.

Matrix. A matrix  of dimension  is a two-dimensional array of real coefficients  where  is the line index,  is the column index. A matrix is usually represented as a table:

A matrix for which  is called a square matrix.

Diagonal. The diagonal of a square  matrix  is the set of coefficients  . A matrix is called diagonal if all its non-diagonal coefficients are zero.

Transpose. The transpose of a  matrix  is a  matrix denoted  with the coefficients defined by  i.e. the coefficients  and  are swapped, which looks like a symmetry with respect to the diagonal:

Symmetry. A square matrix is symmetric if it is equal to its transpose, i.e.  . This is equivalent to having  for any  and  . A property of diagonal matrices is that they are symmetric.

Scalar multiplication. A  matrix  times a real scalar  is defined as the  matrix  with coefficients  .

Matrix sum. The sum of two  matrices  and  is defined as the  matrix  with coefficients  . It is easy to see that the sum and scalar multiplication define a vector space structure on the set of matrices (the sum is associative and its neutral element is the zero matrix, with all coefficients set to zero).

Matrix product. The product between an  matrix  and a  matrix is defined as the  matrix  with coefficients  given by

The product is not defined if the number of columns in  is not the same as the number of lines in  . The product is not commutative in general. The neutral element of the product is the identity matrix  defined as the diagonal matrix with values 1 on the diagonal, and the suitable dimension. If  the product can be generalized to matrix times vector  by identifying the right-hand term of the product with the column  of vector coordinates in a suitable basis; then the multiplication (on the left) of a vector  by a matrix  can be identified to a linear application from  to  . Likewise, matrices can be identified with scalars.

Matrix inverse. A square  matrix  is called invertible if there exist an matrix denoted  and called inverse of  , such that 

Trace. The trace of a square  matrix  is defined as the scalar  which is the sum of the diagonal coefficients.

Useful properties.

(ABC are assumed to be such that the operations below have a meaning)

The transposition is linear

Transpose of a product

Inverse of a product

Inverse of a transpose

Associativity of the product

Diagonal matrices: their products and inverses are diagonal, with coefficients given respectively by the products and inverses of the diagonals of the operands.

Symmetric matrices: the symmetry is conserved by scalar multiplication, sum and inversion, but not by the product (in general).

The trace is linear

Trace of a transpose

Trace of a product

Trace and basis change , i.e. the trace is an intrinsic property of the linear application represented by  .

Positive definite matrices. A symmetric matrix  is defined to be positive definite if, for any vector  , the scalar  unless  . Positive definite matrices have real positive eigenvalues, and their positive definiteness is conserved through inversion.


TJ Morris, Author, Speaker, Entrepreneur, Radio Host
Syndicated Columnist/Investigative Reporter
ACO Ascension Center Organization Founder 1990-92
ACE Nonprofit Inc Founding Director April 2013
ACE Folklife Founder

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