Sum A+cB of invertible matrices noninvertible?

In summary, given two invertible square matrices of the same size with complex entries, it is possible to find a complex scalar c such that when added to one of the matrices, the resulting matrix is non-invertible. This can be proven by constructing c using the hint that a non-invertible matrix has at least one zero eigenvalue.
  • #1
Grothard
29
0
If A and B are both invertible square matrices of the same size with complex entries, there exists a complex scalar c such that A+cB is noninvertible.

I know this to be true, but I can't prove it. I tried working with determinants, but a specific selection of c can only get rid of one entry in A+cB, which is not enough since the matrices do not have to be triangular. How would one go about proving this theorem?
 
Physics news on Phys.org
  • #2
Prove this theorem by construction. Hint: A non-invertible matrix has at least one zero eigenvalue.
 
  • #3
So c = (-λ_a_k)/(-λ_b_k), thanks!
 

FAQ: Sum A+cB of invertible matrices noninvertible?

What is the definition of a noninvertible matrix?

A noninvertible matrix, also known as a singular matrix, is a square matrix that does not have an inverse. This means that it cannot be multiplied by another matrix to produce the identity matrix.

How can you tell if a matrix is noninvertible?

A matrix is noninvertible if its determinant is equal to 0. This means that there is no solution to the equation Ax = b, where A is the noninvertible matrix and b is a vector of constants. In other words, the columns of a noninvertible matrix are linearly dependent.

What is the significance of a noninvertible matrix in linear algebra?

A noninvertible matrix has no inverse, which means that it cannot be used to solve certain systems of linear equations. It also has no eigenvalues, and therefore no eigenvectors. This can make it difficult to analyze and manipulate in linear algebra calculations.

Can a sum of invertible matrices be noninvertible?

Yes, a sum of invertible matrices can result in a noninvertible matrix. This can happen if the matrices do not have the same size or if their sum is not invertible.

How can we handle the presence of noninvertible matrices in calculations?

There are a few ways to handle noninvertible matrices in calculations. One approach is to remove the noninvertible matrix from the calculation entirely. Another option is to use a pseudoinverse, which is a generalization of the inverse for noninvertible matrices. Additionally, numerical methods can be used to approximate the inverse of a noninvertible matrix.

Similar threads

I Spectral theorem for Hermitian matrices-- special cases
Replies
2
Views
2K
I Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form
Replies
7
Views
1K
Difference symmetric matrices vector space and hermitian over R
Replies
1
Views
3K
I Is a symmetric matrix with positive eigenvalues always real?
Replies
8
Views
2K
A The meaning of ##(ict, x_1,x_2,x_3)##
Replies
4
Views
563
Some questions about invertibility of matrix products
Replies
2
Views
2K
MHB Invertible Matrices: Why These Statements Are Not Correct
Replies
1
Views
2K
Are G & H Isomorphic? Prove Answer | Group of 2x2 Matrices
Replies
4
Views
4K
MHB Verifying Answers to True/False Questions about Matrices
Replies
5
Views
2K
Can a matrix of linear forms always be written as the sum of rank one matrices?
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top