Eigenvalue, Eigenvector, Matrix Operations, Quotient Spaces

What is the Characteristic Polynomial used for finding eigenvalues?

The Characteristic Polynomial of a square matrix A is given by p(λ)=det(A−λI), where I is the identity matrix and λ is the eigenvalue. The roots of this polynomial are the eigenvalues of A.

What is the Algebraic Multiplicity of an eigenvalue?

The Algebraic Multiplicity of an eigenvalue λ is the number of times λ appears as a root of the characteristic polynomial. It represents how many times λ is repeated as an eigenvalue.

What is the Geometric Multiplicity of an eigenvalue?

The Geometric Multiplicity of an eigenvalue λ is the number of linearly independent eigenvectors associated with λ. It is always less than or equal to the algebraic multiplicity.

What is the Eigenspace of an eigenvalue?

The Eigenspace corresponding to an eigenvalue λ of a matrix A is the set of all eigenvectors associated with λ, along with the zero vector. It forms a subspace of the vector space.

What does it mean for eigenvectors to be linearly independent?

Eigenvectors are linearly independent if no eigenvector in the set can be written as a linear combination of the others. This is crucial for diagonalizing a matrix, as a set of linearly independent eigenvectors forms a basis for the vector space.

What is the condition for a matrix to be diagonalizable using eigenvectors?

A matrix A is diagonalizable if it has a full set of linearly independent eigenvectors. This is possible when the geometric multiplicity of each eigenvalue is equal to its algebraic multiplicity.

What is the Determinant of a matrix, and how is it related to eigenvalues?

The Determinant of a square matrix A is the product of its eigenvalues, each raised to the power of its algebraic multiplicity. If A is singular (non-invertible), then at least one eigenvalue is zero, making the determinant zero.

What is the Rank of a matrix, and how does it relate to eigenvalues?

The Rank of a matrix A is the dimension of its column space (or row space). It is related to eigenvalues in the sense that the rank of a matrix is the number of non-zero eigenvalues, while the nullity (dimension of the null space) is the number of zero eigenvalues.

What is the difference between row operations and column operations in matrices?

Row operations involve multiplying a row by a scalar, adding a scalar multiple of one row to another, or swapping rows. Column operations are similar but applied to columns. While row operations are used for Gaussian elimination, column operations are less commonly used and generally don't affect the row space directly.

How do you compute the Inverse of a matrix using elementary row operations?

To compute the inverse of a matrix A, augment A with the identity matrix I. Use elementary row operations to transform A into I, and the augmented portion will become A⁻¹, provided A is invertible.

What is the relationship between a Quotient Space and its Subspace?

A quotient space V/W is the set of cosets of the subspace W in the vector space V. Each coset is a vector v+W where v∈V. The quotient space is a vector space, and its dimension is given by dim(V)−dim(W).

How do you compute the dimension of a quotient space V/W?

The dimension of a quotient space V/W is the difference in dimensions of V and W, given by:

dim(V/W) = dim(V) − dim(W)

What is the Projection of a vector onto a subspace, and how is it related to quotient spaces?

The Projection of a vector v onto a subspace W is the vector in W closest to v. The difference between v and its projection lies in the quotient space V/W, as it represents the part of v orthogonal to W.

What is the concept of Coset Representatives in quotient spaces?

In a quotient space V/W, the coset representative is any vector v∈V that represents the coset v+W. Different choices of representatives may correspond to different cosets, but each coset has one representative.

What is the Jordan Canonical Form of a matrix?

The Jordan Canonical Form of a matrix A is a block diagonal matrix where each block corresponds to an eigenvalue, and each block can be either a single eigenvalue or a Jordan block for generalized eigenvectors. It provides a way to simplify the matrix while preserving its eigenvalues and eigenvectors.

What is the Cayley-Hamilton Theorem?

The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic polynomial. In other words, if p_A(λ) is the characteristic polynomial of a matrix A, then p_A(A)=0.