MATRIKS RUANG VEKTOR | VEKTOR EIGEN DAN BASIS RUANG EIGEN

Summary

In this video, the process of determining eigenvectors and eigenspace bases for matrices is explained in detail. The instructor demonstrates how to calculate eigenvalues and solve systems of linear equations (SPL) using row reduction techniques to find the corresponding eigenvectors. Through practical examples, including both 2x2 and 3x3 matrices, the video shows step-by-step how to derive eigenvectors, verify them, and determine the eigenbasis. The key focus is on solving homogeneous linear systems and understanding the relationship between matrices, eigenvalues, and their associated eigenspaces.

Takeaways

• 😀 Eigenvalues are critical in determining eigenvectors and eigenspaces for matrices.
• 😀 The first step in finding eigenvectors is to subtract the eigenvalue multiplied by the identity matrix from the original matrix.
• 😀 For each eigenvalue, a system of linear equations (SPL) is set up to solve for the eigenvectors.
• 😀 The solution to the system provides the eigenvectors, which can be represented with a parameter (e.g., t) for free variables.
• 😀 The basis of an eigenspace is the set of eigenvectors corresponding to a specific eigenvalue.
• 😀 To verify that a vector is indeed an eigenvector, it can be multiplied by the original matrix, and the result should be the eigenvalue times the eigenvector.
• 😀 For a 2x2 matrix, solving the system involves row operations to reduce the coefficient matrix to row echelon form.
• 😀 For an eigenvalue of 2, the eigenvector is (1/2, 1), which forms the basis for the eigenspace.
• 😀 For an eigenvalue of 3, the eigenvector is (1, 1), which forms the basis for its eigenspace.
• 😀 For 3x3 matrices, the process of finding eigenvectors and eigenspaces is the same, with more complex row operations involved.
• 😀 When solving systems for higher-dimensional matrices, understanding row operations and the concepts of eigenvectors and eigenspaces remains essential.

Q & A

What is the primary objective of this video?

The primary objective of the video is to explain how to determine the eigenvectors and eigenbases for a matrix, building upon the concept of eigenvalues from the previous video.

How are eigenvectors and eigenbases related?

Eigenvectors are vectors that satisfy the equation A*v = λ*v, where A is the matrix and λ is the eigenvalue. The eigenbasis refers to the set of eigenvectors that span the eigenspace corresponding to a particular eigenvalue.

What is the first step in finding the eigenvector for λ = 2?

The first step is to substitute λ = 2 into the matrix A - λ*I, where I is the identity matrix. This yields a new matrix that is used to form a system of linear equations.

How is the system of linear equations solved?

The system is solved by using elementary row operations on the coefficient matrix until it is in row echelon form, which simplifies the process of finding the solutions for the variables.

What does the row operation result in for λ = 2?

For λ = 2, the row operations result in a system where the first row becomes '1 -1/2' and the second row is all zeros. This allows the solution to be parameterized by a free variable t.

What is the eigenvector for λ = 2?

The eigenvector corresponding to λ = 2 is [1/2, 1], which is the solution to the system of equations, expressed in terms of the parameter t.

How is the eigenvector for λ = 3 calculated?

For λ = 3, similar steps are followed: substitute λ = 3 into A - λ*I, perform row operations, and solve the system. This gives the eigenvector [1, 1].

What is the eigenbasis for λ = 3?

The eigenbasis for λ = 3 is the set containing the eigenvector [1, 1]. This represents the set of all eigenvectors corresponding to λ = 3.

What is the process for verifying if a vector is indeed an eigenvector?

To verify if a vector is an eigenvector, multiply the matrix A by the vector and check if the result is equal to λ times the vector, where λ is the eigenvalue.

How does the method change when calculating eigenvectors for larger matrices, like 3x3 matrices?

The method remains largely the same for larger matrices, where the matrix A - λ*I is formed, a system of linear equations is solved, and the eigenvectors and eigenbases are found by row reducing the coefficient matrix.