Eigenvalues and Eigenvectors Concepts

The video tutorial explores the concept of eigenvectors and eigenvalues, focusing on determining if a given vector is an eigenvector for specific matrices. The process involves calculating the product of each matrix with the vector and checking if the result is a scalar multiple of the vector. The tutorial evaluates several matrices, identifying which ones have the vector as an eigenvector with a non-zero eigenvalue. The video concludes with a summary of findings, highlighting matrices that meet the criteria.

What is the main goal when determining if a vector is an eigenvector for a given matrix?

• To see if the matrix times the vector equals a scalar multiple of the vector

• To determine if the vector is orthogonal to the matrix

• To check if the matrix is invertible

• To find if the vector is a zero vector

What is the result of the product of the first matrix and the vector negative 1 1?

• The vector negative 1 0

• The vector 1 1

• The zero vector

• A scalar multiple of the vector negative 1 1

For the second matrix, what is the eigenvalue when the vector negative 1 1 is an eigenvector?

• Two

• Negative one

• One

• Zero

Why is the third matrix not selected as having the vector negative 1 1 as an eigenvector with a non-zero eigenvalue?

• The product is not a scalar multiple of the vector

• The product results in the zero vector

• The matrix is not square

• The eigenvalue is negative

What is the eigenvalue for the matrix where the vector 1 negative 1 is a scalar multiple of negative 1 1?

• Zero

• One

• Negative one

• Two

Which of the following matrices results in the vector zero one, indicating it is not an eigenvector?

• Matrix one two zero one

• Matrix two one one two

• Matrix two two zero one

• Matrix one one zero zero

What is the significance of a zero eigenvalue in the context of this problem?

• It indicates the vector is an eigenvector

• It means the vector is not an eigenvector with a non-zero eigenvalue

• It shows the matrix is singular

• It implies the vector is orthogonal to the matrix

How do you determine if a vector is a scalar multiple of another vector?

• By checking if their dot product is zero

• By confirming they are orthogonal

• By verifying they have the same magnitude

• By ensuring one vector is a constant times the other

What is the result of the product of the last matrix and the vector negative 1 1?

• The vector one zero

• The vector negative one one

• The zero vector

• The vector zero one

Which matrix is selected as having the vector negative 1 1 as an eigenvector with a non-zero eigenvalue?

• Matrix one two zero one

• Matrix two one one two

• Matrix two two zero one

• Matrix one one zero zero