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Matrices- Properties of Eigenvalues- 2 Video Lecture - Physics

FAQs on Matrices- Properties of Eigenvalues- 2 Video Lecture - Physics

1. What are eigenvalues and eigenvectors of a matrix?

Ans. Eigenvalues and eigenvectors are properties of a square matrix. Eigenvalues are scalar values that represent the scaling factor of the eigenvectors when the matrix is multiplied by them. Eigenvectors are non-zero vectors that remain in the same direction after being multiplied by the matrix.

2. How can eigenvalues be calculated for a given matrix?

Ans. To calculate the eigenvalues of a matrix, we need to solve the characteristic equation. The characteristic equation is obtained by subtracting the identity matrix multiplied by a scalar λ from the given matrix, and then finding the determinant of the resulting matrix. The values of λ that satisfy the characteristic equation are the eigenvalues of the matrix.

3. What are the properties of eigenvalues?

Ans. Eigenvalues have several important properties. Some of these properties include: - The sum of the eigenvalues is equal to the trace of the matrix. - The product of the eigenvalues is equal to the determinant of the matrix. - The eigenvalues are unchanged under similarity transformation. - The eigenvalues of a diagonal matrix are the diagonal entries themselves. - The eigenvalues of a triangular matrix are the diagonal entries.

4. How do eigenvalues relate to the diagonalizability of a matrix?

Ans. A matrix is said to be diagonalizable if it has a full set of linearly independent eigenvectors. The eigenvalues of a diagonalizable matrix correspond to the diagonal entries of the diagonalized form of the matrix. In other words, if a matrix has distinct eigenvalues and is diagonalizable, it can be transformed into a diagonal matrix by using the eigenvectors as columns of a transformation matrix.

5. Can a matrix have complex eigenvalues?

Ans. Yes, a matrix can have complex eigenvalues. The eigenvalues of a matrix can be real or complex, depending on the matrix itself. Complex eigenvalues often arise when dealing with matrices that involve complex numbers or have complex coefficients. The presence of complex eigenvalues indicates the existence of complex eigenvectors as well.

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Mar 07, 2025 Last updated

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