Question

The eigen values of a triangular matrix are ...

Short answer

The eigenvalues of a triangular matrix are the elements on its main diagonal.

Step-by-step solution

Understand the Matrix Type

A triangular matrix can be either upper triangular, lower triangular, or a diagonal matrix. A matrix is upper triangular if all the elements below its main diagonal are zero. Similarly, a matrix is lower triangular if all the elements above its main diagonal are zero.

Identify the Main Diagonal

In a triangular matrix, the main diagonal consists of elements from the top-left to the bottom-right corner of the matrix. These elements are along the diagonal where the row and column indices are equal.

Recall the Eigenvalue Property for Triangular Matrices

For any triangular matrix (whether upper or lower triangular), the eigenvalues are given by its diagonal elements. This is because the determinant, which is used to calculate eigenvalues, simplifies to the product of the diagonal elements in a triangular matrix.

State the Result

The eigenvalues of a triangular matrix are exactly the elements on its main diagonal. Thus, if the main diagonal elements are \(d_1, d_2, \ldots, d_n\), then these are the eigenvalues of the matrix.

Key concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of matrix where all the elements below the main diagonal are zeros. In other words, if you look at the matrix as a triangle, all the values in the bottom corner are zeros.

• Only the elements on or above the main diagonal can be non-zero.
• It's often represented as follows: for a matrix \( A \), the elements \( a_{ij} \) where \( i > j \) are zero.

Upper triangular matrices have a neat property that simplifies many calculations, especially when finding determinant or eigenvalues. It's because the non-zero elements are concentrated in a specific part of the matrix.

Lower Triangular Matrix

Contrary to upper triangular matrices, lower triangular matrices have all their non-zero elements on or below the main diagonal.

• In this case, the elements on the upper side of the main diagonal are zeros.
• A general form of a lower triangular matrix would have \( a_{ij} = 0 \) for all \( i < j \).

Lower triangular matrices also make computations easier, similar to upper triangular matrices. Especially when calculating the determinant or simplifying systems of linear equations. Their structure can quite effectively be leveraged in various mathematical and computational processes.

Diagonal Matrix

A diagonal matrix is a step further in specialization from triangular matrices. Here, only the elements on the main diagonal are non-zero; all other elements in the matrix are zeros.

• This means only \( a_{ii} \) (where row and column indices are equal) are non-zero.
• Diagonal matrices are the simplest among triangular matrices.

These matrices are incredibly simple to work with. For instance, the determinant of a diagonal matrix is simply the product of its diagonal elements, and its eigenvalues are directly the diagonal elements themselves.

Main Diagonal

The main diagonal in a square matrix extends from the top-left corner to the bottom-right corner.

• These are the elements \( a_{ii} \), where the row and column indices are identical.
• In triangular or diagonal matrices, these diagonal elements play a crucial role in many calculations.

The importance of the main diagonal is seen in calculating eigenvalues. Especially in triangular matrices where these diagonal values are directly related to the eigenvalues of the matrix.

Matrix Determinant

The determinant of a square matrix is a scalar value that offers insight into the matrix properties, like invertibility.

• For any square matrix \( A \), the determinant is denoted by \( \text{det}(A) \).
• In triangular matrices (both upper and lower), the determinant is simply the product of the main diagonal elements.

This simplification allows for fast determinant calculation in triangular matrices without needing complex transformations. The matrix determinant can thus reveal important characteristics such as the matrix having full rank or if it has eigenvalues of zero.