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Chapter 6: Problem 18

Let \(A\) be an \(n \times n\) matrix and let \(\lambda\) be an eigenvalue of \(A .\) If \(A-\lambda I\) has rank \(k,\) what is the dimension of the eigenspace corresponding to \(\lambda ?\) Explain.

Short Answer

Expert verified
The dimension of the eigenspace corresponding to the eigenvalue \(\lambda\) is equal to \(n-k\), where \(n\) is the number of columns of the matrix \(A\), and \(k\) is the rank of the matrix \((A-\lambda I)\).

Step by step solution

01

Find the null space of (A鈭捨籌)

To find the dimension of the eigenspace corresponding to the eigenvalue \(\lambda\), we need to find the null space of the matrix \((A-\lambda I)\). The null space is the set of all vectors \(x\), such that \((A-\lambda I)x=0\). Since we know the rank \((A-\lambda I)=k\), we can use the Rank-Nullity theorem.
02

Use the Rank-Nullity theorem

The Rank-Nullity theorem states that for any matrix \(M\), the sum of its rank and the dimension of its null space (nullity) is equal to the number of columns of the matrix. In this case, the matrix \(M\) is \((A-\lambda I)\), which is an \(n\times n\) matrix. We are given that rank\((A-\lambda I)=k\). Therefore, we can use the Rank-Nullity theorem to find the dimension of the eigenspace (nullity). The Rank-Nullity theorem is given by: \(rank(M)+nullity(M) = n\) Substitute the given values into the equation: \(k+nullity(A-\lambda I) = n\)
03

Solve for the dimension of the eigenspace

We now need to solve the equation for the dimension of the eigenspace, which is the nullity of \((A-\lambda I)\): \(k+nullity(A-\lambda I) = n\) Rearrange the equation to solve for the nullity of \((A-\lambda I)\): \(nullity(A-\lambda I) = n-k\) Therefore, the dimension of the eigenspace corresponding to the eigenvalue \(\lambda\) is equal to \(n-k\).

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Most popular questions from this chapter

Let \(A\) be a diagonalizable matrix whose eigenvalues are all either 1 or \(-1 .\) Show that \(A^{-1}=A\)

Let \(T\) be an upper triangular matrix with distinct diagonal entries (i.e., \(t_{i i} \neq t_{j j}\) whenever \(i \neq j\) ). Show that there is an upper triangular matrix \(R\) that diagonalizes \(T\)

Let \(A\) be a symmetric positive definite matrix. Show that the diagonal elements of \(A\) must all be positive.

Show that if \(A\) is a normal matrix, then each of the following matrices must also be normal: (a) \(A^{H}\) (b) \(I+A\) (c) \(A^{2}\)

Let \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\\}\) be an orthonormal basis for \(\mathbb{C}^{2},\) and let \(\mathbf{z}=(4+2 i) \mathbf{u}_{1}+(6-5 i) \mathbf{u}_{2}\) (a) What are the values of \(\mathbf{u}_{1}^{H} \mathbf{z}, \mathbf{z}^{H} \mathbf{u}_{1}, \mathbf{u}_{2}^{H} \mathbf{z},\) and \(\mathbf{z}^{H} \mathbf{u}_{2} ?\) (b) Determine the value of \(\|\mathbf{z}\|\)
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