There is a special importance to the product of a matrix and a vector in linear algebra. Consider the product of a 3 x 2 matrix A and a 2 x 1 vector x producing a 3 x 1 vector y:

(C)

(R)

It is useful to consider two views of the preceding matrix-vector product, namely, the column picture (C) and the row picture (R). In the column picture, the product can be seen as a linear combination of the column vectors of the matrix, whereas the row picture can be thought of as the dot products of the rows of the matrix with the vector

The product of a matrix with its inverse is the Identity matrix. Thus:

The matrix inverse, if it exists, can be used to solve a system of simultaneous equations represented by the preceding vector-matrix product equation. Consider a system of equations:

x1 + 2x2 = 3

3x1 + 9x2 = 21

This can be expressed as an equation involving the matrix-vector product:

We can solve for the variables x1 and x2 by multiplying both sides by the matrix inverse:

The matrix inverse can be calculated by different methods. The reader is advised to view Prof. Strang's MIT lecture: bit.ly/10vmKcL.

Matrices can be decomposed to factors that can give us valuable insight into the transformation that the matrix represents. Eigenvalues and eigenvectors are obtained as the result of eigendecomposition. For a given square matrix A, an eigenvector is a non-zero vector that is transformed into a scaled version of itself when multiplied by the matrix. The scalar multiplier is the eigenvalue. All scalar multiples of an eigenvector are also eigenvectors:

A v = λ v

In the preceding example, v is an eigenvector and λ is the eigenvalue.

The eigenvalue equation of matrix A is given by:

(A — λ I)v = 0

The non-zero solution for the eigenvalues is given by the roots of the characteristic polynomial equation of degree n represented by the determinant:

The eigenvectors can then be found by solving for v in Av = λ v.

Some matrices, called diagonalizable matrices, can be built entirely from their eigenvectors and eigenvalues. If Λ is the diagonal matrix with the eigenvalues of matrix A on its principal diagonal, and Q is the matrix whose columns are the eigenvectors of A:

Then A = Q Λ Q-1.

If a matrix has only positive eigenvalues, it is called a positive definite matrix. If the eigenvalues are positive or zero, the matrix is called a positive semi-definite matrix. With positive definite matrices, it is true that:

xT Ax ≥ 0