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Linear Algebra in MATLAB: Defining Matrices, Matrix Operations, and Solving Linear Systems, Lecture notes of Linear Algebra

An introduction to linear algebra in MATLAB, covering the basics of defining and referring to matrices, matrix operations and arithmetic, and solving systems of linear equations. It includes examples and explanations of defining matrices, matrix addition, multiplication, and the computation of determinants, transposes, powers, inverses, and eigenvalues.

What you will learn

  • What is the difference between matrix addition and multiplication in MATLAB?
  • How do you find the determinant, inverse, and eigenvalues of a matrix in MATLAB?
  • How do you define a matrix in MATLAB?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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Linear Algebra in MATLAB
As suggested by its name, MATLAB (MATrix LABoratory) was originally developed as a
set of numerical algorithms for manipulating matrices.
Defining and Referring to a Matrix
First, we have already learned how to define 1 ×nmatrices, which are simply row vectors.
For example, we can define the 1 ×5 matrix
~v = [ 0.2.4.6.8]
with
>>v=[0 .2 .4 .6 .8]
v =
0 0.2000 0.4000 0.6000 0.8000
>>v(1)
ans =
0
>>v(4)
ans =
0.6000
We also see in this example that we can refer to the entries in vwith v(1), v(2), etc.
In defining a more general matrix, we proceed similarly by defining a series of row vectors,
ending each with a semicolon.
Example 1. Define the matrix
A=
123
456
789
in MATLAB.
We accomplish this the following MATLAB code, in which we also refer to the entries
a21 and a12.
>>A=[1 2 3;4 5 6;7 8 9]
A =
1 2 3
4 5 6
7 8 9
>>A(1,2)
ans =
2
>>A(2,1)
ans =
4
1
pf3
pf4
pf5

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Linear Algebra in MATLAB

As suggested by its name, MATLAB (MATrix LABoratory) was originally developed as a set of numerical algorithms for manipulating matrices.

Defining and Referring to a Matrix

First, we have already learned how to define 1 × n matrices, which are simply row vectors. For example, we can define the 1 × 5 matrix

~v = [ 0. 2. 4. 6 .8 ]

with

v=[0 .2 .4 .6 .8] v = 0 0.2000 0.4000 0.6000 0. v(1) ans = 0 v(4) ans =

We also see in this example that we can refer to the entries in v with v(1), v(2), etc. In defining a more general matrix, we proceed similarly by defining a series of row vectors, ending each with a semicolon.

Example 1. Define the matrix

A =

in MATLAB. We accomplish this the following MATLAB code, in which we also refer to the entries a 21 and a 12.

A=[1 2 3;4 5 6;7 8 9] A = 1 2 3 4 5 6 7 8 9 A(1,2) ans = 2 A(2,1) ans = 4

In general, we refer to the entry aij with A(i, j). We can refer to an entire column or an entire row by replacing the index for the row or column with a colon. In the following code, we refer to the first column of A and to the third row.

A(:,1) ans = 1 4 7 A(3,:) ans = 7 8 9

Matrix Operations and Arithmetic

MATLAB adds and multiplies matrices according to the standard rules.

Example 2. For the matrices

A =

 (^) and B =

compute A + B, AB, and BA. Assuming A has already been defined in Example 1, we use

B=[0 1 3;2 4 1;6 1 8] B = 0 1 3 2 4 1 6 1 8 A+B ans =

1 3 6 6 9 7 13 9 17

AB ans = 22 12 29 46 30 65 70 48 101 BA ans = 25 29 33 25 32 39 66 81 96

In addition to returning the eigenvalues of a matrix, the command eig will also return the associated eigenvectors. For a given square matrix A, the MATLAB command

[P, D] = eig(A)

will return a diagonal matrix D with the eigenvalues of A on the diagonal and a matrix P whose columns are made up of the eigenvectors of A. More precisely, the first column of P will be the eigenvector associated with d 11 , the second column of P will be the eigenvector associated with d 22 etc.

Example 4. Compute the eigenvalues and eigenvectors for the matrix B from Example 3. We use

[P,D]=eig(B) P = -0.8480 0.2959 -0. A(2,1) ans = 4 In general, we refer to the entry aij with A(i, j). We can refer to an entire column or an entire row by replacing the index for the row or column with a colon. In the following code, we refer to the first column of A and to the third row. >>A(:,1) ans = 1 4 7 >>A(3,:) ans = 7 8 9 △ ## Matrix Operations and Arithmetic MATLAB adds and multiplies matrices according to the standard rules. Example 2. For the matrices ### A =  (^) and B = compute A + B, AB, and BA. Assuming A has already been defined in Example 1, we use >>B=[0 1 3;2 4 1;6 1 8] B = 0 1 3 2 4 1 6 1 8 >>A+B ans = 1 3 6 6 9 7 13 9 17 >>AB ans = 22 12 29 46 30 65 70 48 101 >>BA ans = 25 29 33 25 32 39 66 81 96 In addition to returning the eigenvalues of a matrix, the command eig will also return the associated eigenvectors. For a given square matrix A, the MATLAB command [P, D] = eig(A) will return a diagonal matrix D with the eigenvalues of A on the diagonal and a matrix P whose columns are made up of the eigenvectors of A. More precisely, the first column of P will be the eigenvector associated with d 11 , the second column of P will be the eigenvector associated with d 22 etc. Example 4. Compute the eigenvalues and eigenvectors for the matrix B from Example 3. We use >>[P,D]=eig(B) P = -0.8480 0.2959 -0. 0.2020 0.2448 -0. 0.4900 0.9233 0. D = -1.9716 0 0 0 10.1881 0 0 0 3.

We see that the eigenvalue–eigenvector pairs of B are

Let’s check that these are correct for the first one. We should have B~v 1 = − 1. 9716 ~v 1. We can check this in MATLAB with the calculations

B*P(:,1) ans =

-0. -0.

D(1,1)*P(:,1) ans =

-0. -0.

Solving Systems of Linear Equations

Example 5. Solve the linear system of equations

x 1 + 2x 2 + 7x 3 = 1 2 x 1 − x 2 = 3 9 x 1 + 5x 2 + 4x 3 = 0.

We begin by defining the coefficients on the left hand side as a matrix and the right hand side as a vector. If

A =

 (^) , ~x =

x 1 x 2 x 3

 (^) and ~b =

then in matrix form this equation is A~x = ~b,

which is solved by ~x = A−^1 ~b.

In order to compute this in MATLAB, we use the following code.

A=[1 2 7;2 -1 0;9 5 4] A = 1 2 7 2 -1 0 9 5 4 b=[1 3 0]’ b = 1 3 0 x=Aˆ(-1)*b x =

-1.

x=A\b x =

-1.

Notice that MATLAB understands multiplication on the left by A−^1 , and interprets division on the left by A the same way. △ If the system of equations we want to solve has no solution or an infinite number of solutions, then A will not be invertible and we won’t be able to proceed as in Example 5. In

In this case, we see immediately that the third equation asserts 0 = 1, which means there is no solution to this system. △

Assignments

  1. For the matrix

A =

compute the determinant, transpose, 3rd power, inverse, eigenvalues and eigenvectors.

  1. For matrix A from Problem 1, let P denote the matrix created from the eigenvectors of A and compute P −^1 AP.

Explain how the result of this calculation is related to the eigenvalues of A.

  1. Find all solutions of the linear system of equations

x 1 + 2x 2 − x 3 + 7x 4 = 2 2 x 1 + x 2 + 9x 3 + 4x 4 = 1 3 x 1 + 11x 2 − x 3 + 7x 4 = 0 4 x 1 − 6 x 2 − x 3 + 1x 4 = − 1.

  1. Find all solutions of the linear system of equations

− 3 x 1 − 2 x 2 + 4x 3 = 0 14 x 1 + 8x 2 − 18 x 3 = 0 4 x 1 + 2x 2 − 5 x 3 = 0.

  1. Find all solutions of the linear system of equations

x 1 + 6x 2 + 4x 3 = 1 2 x 1 + 4x 2 − 1 x 3 = 0 −x 1 + 2x 2 + 5x 3 = 0.