Identity matrix

Definition

An identity matrix is a square matrix in which all the elements of the principal diagonal are ones, and all other elements are zeros. This matrix acts as a multiplicative identity in linear algebra, meaning that when any matrix is multiplied by the identity matrix, the result is the original matrix. It plays a crucial role in solving eigenvalue problems and in determining characteristics of linear transformations.

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 (number of rows and columns).
2. For any square matrix $$A$$ of size $$n \times n$$, multiplying it by the identity matrix $$I_n$$ gives back the original matrix: $$A \cdot I_n = A$$.
3. The identity matrix serves as the equivalent of the number 1 in matrix multiplication, allowing for simplification in algebraic manipulations.
4. In eigenvalue problems, the identity matrix is used to formulate the characteristic equation by subtracting $$\lambda I$$ from the original matrix and calculating the determinant.
5. The presence of an identity matrix is essential for defining invertible matrices; a square matrix is invertible if it can be multiplied by another matrix to yield the identity matrix.

Review Questions

• How does an identity matrix facilitate the process of solving eigenvalue problems?
  • An identity matrix allows for the formulation of the characteristic equation in eigenvalue problems. By subtracting $$\lambda I$$ from a given square matrix $$A$$, we create a new matrix whose determinant can be set to zero to find the eigenvalues. This step is crucial because it simplifies finding eigenvalues, ultimately leading to solving for eigenvectors as well.
• Explain how multiplying any square matrix by an identity matrix affects its properties and provide an example.
  • Multiplying any square matrix by an identity matrix retains all properties of that original matrix, effectively demonstrating that the identity acts like '1' in traditional arithmetic. For example, if we have a 2x2 matrix $$A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$$ and multiply it by the 2x2 identity matrix $$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$, we get back $$A$$: $$A \cdot I_2 = A$$. This property is fundamental in ensuring that transformations remain consistent.
• Evaluate the implications of having an identity matrix within a system of linear equations and how it relates to invertible matrices.
  • In a system of linear equations, the presence of an identity matrix indicates that there exists an inverse for a given coefficient matrix, meaning the system can be solved uniquely. If we can express our system in terms of $$AX = B$$ and manipulate it to yield an identity matrix through row operations or inverse calculations, we establish that solutions are attainable. This also highlights that for a square coefficient matrix to be invertible, it must be possible to manipulate it into an identity form through elementary row operations or multiplication with its inverse.

Related terms

Eigenvalues: Eigenvalues are scalars associated with a linear transformation represented by a matrix, indicating how much the transformation stretches or compresses vectors.

Eigenvectors: Eigenvectors are non-zero vectors that change only in scale when a linear transformation is applied, representing directions that remain invariant under the transformation.

Characteristic polynomial: The characteristic polynomial is a polynomial which is derived from a square matrix, used to find its eigenvalues by setting the determinant of the matrix minus lambda times the identity matrix to zero.