Identity matrix

Definition

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere, acting as the multiplicative identity in matrix multiplication. When a matrix is multiplied by the identity matrix, it remains unchanged, which establishes its foundational role in linear transformations. The identity matrix is crucial for understanding invertible transformations, eigenvalues, and eigenvectors, as well as characteristics of positive definite matrices.

5 Must Know Facts For Your Next Test

1. The identity matrix is denoted as $$I_n$$, where $$n$$ indicates the size of the matrix, such as $$I_2$$ for a 2x2 identity matrix.
2. In any dimension, multiplying a matrix by an identity matrix leaves the original matrix unchanged, demonstrating its role as the multiplicative identity.
3. The identity matrix serves as a reference point in linear algebra, simplifying many operations such as finding inverses or solving systems of equations.
4. The eigenvalues of the identity matrix are all equal to one, indicating that every vector is an eigenvector associated with this value.
5. In positive definite matrices, the identity matrix can be seen as a basic building block, ensuring that all eigenvalues are positive.

Review Questions

• How does the identity matrix relate to the concept of invertible linear transformations?
  • The identity matrix plays a vital role in invertible linear transformations as it represents the transformation that leaves vectors unchanged. When applying an invertible transformation to any vector and then multiplying by the inverse transformation, you essentially return to your original vector, which can be mathematically expressed using the identity matrix. This shows that if a transformation is invertible, there exists an inverse such that their product results in the identity matrix.
• Discuss how eigenvalues and eigenvectors of the identity matrix reflect its properties compared to other matrices.
  • The eigenvalues of the identity matrix are all equal to one, meaning every vector in the space is an eigenvector corresponding to this eigenvalue. This characteristic differentiates it from other matrices where eigenvalues may vary widely. In contrast to non-identity matrices that can compress or stretch space based on their eigenvalues, the identity matrix preserves all vectors' lengths and directions.
• Evaluate the significance of the identity matrix in determining whether a given square matrix is positive definite.
  • The presence of an identity matrix as part of a positive definite matrix indicates that all eigenvalues are positive. When assessing whether a given square matrix is positive definite, examining its relationship with the identity matrix can clarify its behavior under linear transformations. If you can show that a square matrix minus some scalar times the identity results in a positive definite form, you establish its positive definiteness directly through its eigenvalues being greater than zero.

Related terms

Inverse Matrix: An inverse matrix is a matrix that, when multiplied with the original matrix, yields the identity matrix, allowing for the solution of linear equations.

Diagonal Matrix:

A diagonal matrix is a square matrix where all off-diagonal elements are zero, and it can include identity matrices as a specific case.

Determinant: The determinant is a scalar value that can be computed from a square matrix, providing important information about the matrix's invertibility and properties.