Understanding Upper Triangular Matrices and Their Singularity

Are upper triangular matrices always singular, or can they be non-singular under certain conditions? This article explores the conditions under which an upper triangular matrix is non-singular, along with a detailed proof and relevant examples. We will also discuss the relationship between the singularity of a matrix and its determinant.

Conditions for Non-Singularity in Upper Triangular Matrices

An upper triangular matrix can be non-singular if and only if all of its diagonal entries are non-zero. This is because the determinant of an upper (or lower) triangular matrix is the product of its diagonal entries. If the determinant is non-zero, the matrix is non-singular; otherwise, it is singular.

Consider the following upper triangular matrix:

[ T begin{pmatrix}3 7 0 8 end{pmatrix} ]

The determinant of this matrix is calculated as:

[ det(T) 3 times 8 - 7 times 0 24 ]

Since the determinant is non-zero, the matrix ( T ) is non-singular and thus invertible.

Proof of Non-Singularity for Triangular Matrices

We will now provide a proof that a triangular matrix is non-singular if and only if all of its diagonal elements are non-zero.

Claim: A triangular matrix is non-singular if and only if all of its diagonal elements are non-zero.

Proof:

First, recall that a matrix is non-singular if and only if its determinant is non-zero. For a triangular matrix ( T ), the determinant can be found by taking the product of its diagonal elements. We will consider this for an upper triangular matrix, although the proof for a lower triangular matrix would be analogous.

For an upper triangular matrix of order ( n ), denote the diagonal elements by ( a_{ii} ). The determinant of ( T ) can be expressed as:

[ det(T) prod_{i1}^{n} a_{ii} ]

By the properties of determinants, the diagonal elements determine the non-zero nature of the determinant. If any diagonal element ( a_{ii} ) is zero, then the determinant is zero, making the matrix singular. Conversely, if all diagonal elements are non-zero, the determinant is non-zero, and the matrix is non-singular.

Explanation with Induction

To further solidify the proof, we can use mathematical induction. We will perform the proof for an upper triangular matrix of order ( n ).

Basis Step (n1): For a ( 1 times 1 ) matrix, the determinant is simply the single entry itself. If this entry is non-zero, the matrix is non-singular. This is trivially true.

Inductive Step: Assume the statement is true for a lower triangular matrix of order ( n-1 ). We need to show that it is true for a lower triangular matrix of order ( n ).

Consider an upper triangular matrix of order ( n ): [ T begin{pmatrix} a_{11} a_{12} cdots a_{1n} 0 a_{22} cdots a_{2n} vdots vdots ddots vdots 0 0 cdots a_{nn} end{pmatrix} ]

Using the expansion of the determinant by minors along the first row, we get:

[ det(T) a_{11} pm a_{12} D_{21} pm a_{13} D_{31} pm cdots pm a_{1n} D_{n1} ]

Here, ( D_{ij} ) is the determinant of the ((n-1) times (n-1)) submatrix obtained by deleting the (i)-th row and (j)-th column from ( T ).

For ( i > 1 ), the corresponding minor ( D_{i1} ) contains a zero in the first row, making ( D_{i1} 0 ). Therefore, all terms except ( a_{11} det(T_{[2:n, 2:n]}) ) are zero, where ( T_{[2:n, 2:n]} ) is the ((n-1) times (n-1)) lower triangular matrix obtained by removing the first row and the first column from ( T ).

By the inductive hypothesis, ( det(T_{[2:n, 2:n]}) eq 0 ) if all diagonal entries of ( T_{[2:n, 2:n]} ) are non-zero. Since ( a_{11} eq 0 ) by the inductive hypothesis, we have ( det(T) eq 0 ).

Thus, ( T ) is non-singular if and only if all diagonal entries are non-zero. This completes the proof by induction.

Conclusion

The key takeaway is that the singularity or non-singularity of an upper triangular matrix depends on the non-zero nature of its diagonal elements. By expanding the determinant through minors, we can verify that the product of non-zero diagonal elements ensures that the matrix is non-singular.

Understanding these concepts is crucial in linear algebra, numerical analysis, and various applications in mathematics and engineering.