Eigenvalues and eigenvectors (eig) in MATLAB - Step-by-Step Execution

Concept Flow - Eigenvalues and eigenvectors (eig)

Input matrix A

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Call eig(A)

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Compute eigenvalues and eigenvectors

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Return eigenvalues (D) and eigenvectors (V)

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Use or display results

Start with a matrix A, call eig(A) to find its eigenvalues and eigenvectors, then use or display these results.

Execution Sample

MATLAB

A = [2 1; 1 2];
[V,D] = eig(A);
disp(D);
disp(V);

This code finds eigenvalues and eigenvectors of matrix A and displays them.

Execution Table

Step	Action	Matrix/Variable	Result/Value
1	Define matrix A	A	[2 1; 1 2]
2	Call eig(A)	eig(A)	Compute eigenvalues and eigenvectors
3	Calculate eigenvalues	D	[1 0; 0 3] (diagonal matrix)
4	Calculate eigenvectors	V	[ -0.7071 0.7071; 0.7071 0.7071 ]
5	Display D	disp(D)	Shows eigenvalues on diagonal
6	Display V	disp(V)	Shows eigenvectors as columns
7	End		All eigenvalues and eigenvectors computed

💡 All eigenvalues and eigenvectors computed and displayed

Variable Tracker

Variable	Start	After eig call	Final
A	[2 1; 1 2]	[2 1; 1 2]	[2 1; 1 2]
D	undefined	[1 0; 0 3]	[1 0; 0 3]
V	undefined	[-0.7071 0.7071; 0.7071 0.7071]	[-0.7071 0.7071; 0.7071 0.7071]

Key Moments - 3 Insights

Why are eigenvalues shown in a diagonal matrix D instead of a simple list?

In the execution_table row 3, eigenvalues are stored in a diagonal matrix D because MATLAB's eig function returns them this way to align with eigenvectors in V.

Why are eigenvectors shown as columns in matrix V?

As shown in row 4, each column of V corresponds to an eigenvector matching the eigenvalue in the same column of D, so columns represent eigenvectors.

What does the eig function actually compute?

From rows 2-4, eig computes values λ and vectors v such that A*v = λ*v, which are the eigenvalues and eigenvectors.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what does matrix D represent?

A Matrix of eigenvectors

B The original matrix A

C A diagonal matrix of eigenvalues

D A matrix of zeros

💡 Hint

Refer to execution_table row 3 where D is described as a diagonal matrix of eigenvalues

At which step are eigenvectors first computed according to the execution_table?

A Step 4

B Step 3

C Step 2

D Step 5

💡 Hint

Check execution_table row 4 where eigenvectors V are calculated

If matrix A was changed to a 3x3 matrix, how would the size of D and V change?

A D would be 3x1 and V would be 3x3

B D and V would both be 3x3 matrices

C D would be 3x3 and V would be 3x1

D D and V would both be 1x3 matrices

💡 Hint

Refer to variable_tracker showing D and V are square matrices matching size of A

Concept Snapshot

eig(A) returns eigenvalues and eigenvectors of matrix A.
Syntax: [V,D] = eig(A)
D is diagonal matrix of eigenvalues.
Columns of V are eigenvectors.
Used to solve A*v = lambda*v.

Full Transcript

We start with a matrix A. When we call eig(A), MATLAB calculates eigenvalues and eigenvectors. The eigenvalues are returned in a diagonal matrix D, where each diagonal element is an eigenvalue. The eigenvectors are returned as columns in matrix V, each corresponding to an eigenvalue in D. We display D and V to see the results. This process helps us understand how matrix A transforms vectors by scaling them along eigenvectors.