Chapter 5MATRIX ALGEBRA: DETEMINANT, REVERSE, EIGENVALUES

The determinant of a matrix is a fundamental concept of linear algebra that provides existence and uniqueness results for linear systems of equations; det A = |A|

LU factorization: A = LU, the determinant is|A| = |L||U| (5.1)

Doolittle method: L = lower triangular matrix with lii = 1 |L| = 1

|A| = |U| = u11u22…unn

Pivoting: each time we apply pivoting we need to change the sign of the determinant

|A| = (-1)mu11u22…unn

Gauss forward elimination with pivoting:

5.1 The Determinant of a Matrix

Procedure for finding the determinant following the elimination method

If A = BC |A| = |B||C| (A, B, C – square matrices) |AT| = |A| If two rows (or columns) are proportional, |A| = 0 The determinant of a triangular matrix equals the product

of its diagonal elements A factor of any row (or column) can be placed before the

determinant

Interchanging two rows (or columns) changes the determinant sign

The properties of the determinant

Definition: A-1 is the inverse of the square matrix A, if(5.5)

I – identity matrix A-1 = X (denote) AX = I

(5.6) LU factorization, Doolittle method: A = LU

LY = I , UX = Y (5.7) Here Y = {yij}, LY = I – lower triangular matrix with unity diagonal eleme

nts, i = 1,2,…,n; so for j from 1 to n

Also, X = {xij} vector, i = 1,2,…,n

5.2 Inverse of a Matrix

Procedure for finding the inverse matrix following the LU factorization method

If Ax = b x = A-1b (A - matrix nxn, b, x – n-dimensional vectors) If AX = B X = A-1B (A – matrix nxn, B,X – matrices nxm) – more often

case. We can write down this system as

Axi = bi (i = 1,2,…,m) xi, bi – vectors, ith rows of matrices X, B, respectively

X = A-1B calculating procedure following the method of LU factorization

Definition: let A be an nxn matrix For some nonzero column vector x it may happen, for some scalar λ

Ax = λx (5.9) Then λ is an eigenvalue of A, x is eigenvector of A, associated with the

eigenvalue λ; the problem of finding eigenvalue or eigenvector – eigen problem

Eq.(5.9) can be written in the form Av = λIv or (A – λI)x = 0 If A is nonsingular matrix inverse exists det A ≠ 0

x = (A – λI)-10 = 0 Hence, not to get zero solution x = 0, (A – λI) must not be nonsingular,

i.e. det A = 0:

(5.10)

(5.11)

5.3 Eigenvalues and Eigenvectors

Eq. (5.11) -nth order algebraic equation with n unknowns (algebraic as well as complex)

Applying some numerical calculation, we can find λ But when n is big, expanding to Eq.(5.11) is not the easy way to solve

this method is not used much Here: Jacobi and QR methods for finding eigenvalues

Jacobi method is the direct method for finding eigenvalues and eigenvectors in case when A is the symmetric matrix

diag(a11,a12,…,ann) – diagonal matrix A ei – unit vector with ith element = 1 and all others = 0

The properties of egenvalues and eigenvectors:1. if x – eigenvector of A, then ax is also eigenvector (a = constant)2. if A – diagonal matrix, then aii are eigenvalues, ei is eigenvector

3. if R, Q are orthogonal matrices, then RQ is orthogonal4. if λ – eigenvalue of A, x – eigenvector of A, R – orthogonal matrix, t

hen λ is eigenvalue of RTAR, RTx is its eigenvector. RTAR is called similarity transformation

5. Eigenvalues of a symmetric matrix are real number

5.3.1 Jacobi method

Jacobi method uses the properties mentioned above A – arbitrary matrix, Ri – orthogonal matrix, Ri

TARi – similarity transformation

(5.12) - diagonal matrix following the 2nd property, the eigenvector is ei

We can write the eq.(5.12) as

Following the 3rd property, R1R2…Rn is orthogonal matrix; the 4th property – it has the same eigenvalues as A

The eigenvector of A is

Then the matrix consisting of xi as columns

Having found X, we find eigenvetors xi

Let’s consider 2-dimensional matrix example

The orthogonal matrix is

C, S – notation for cosθ, sinθ respectively We need to choose θ so that the above matrix becomes diagonal

If a11≠a22, then

If a11=a22, then θ=π/4 With this θ, RTAR is diagonal matrix, its diagonal elements are eigenval

ues of A, and R the eigenvecotrs matrix

If A is nxn matrix, aij – its non-diagonal elements with the largest absolute value, then orthogonal matrix Rk and θ:

(5.15)

(5.16)

Then if we calculate RTkARk, after transformation elements a*ij

(5.17)

Then again repeat the process, selecting the largest absolute valued non-diagonal elements and reducing them to zero

Convergence condition:(5.18)

Jacobi method calculation procedure

Jacobi method model (1)

Jacobi model (2)

To find the eigenvalues and eigenvectors of a real matrix A, three methods are combined:

Pretreatment - Householder transformation Calculation of eigenvalues - QR method Calculation of eigenvectors - inverse power method

5.3.2 QR method

Series of orthogonal transformations Ak+1= PkTAkPk

For k = 1, … , n-2, starting from initial matrix A = A1 and applying the similarity transformation until we get three-diagonal matrix An-1

A three-diagonal matrix

Matrix Pk, n-dimensional vector uk

(1) Householder transformation

Matrix Pk, n-dimensional vector uk

ukTuk = 1 = I. Pk – symmetric matrix and through

it is also orthogonal

Orthogonal matrix satisfies the following statement:

If we have two column-vectors x, y, x≠y and ||x|| = ||y||, and if we assign

then

In case k = 1

Here

Since ||b1|| = ||s1e1||

Jacobi method model (1)

Jacobi model (2)