# Given some of the eigenvalues and corresponding eigenvectors reconstruct the entire matrix.

Given a symmetric matrix $$A$$ that has eigenvalues $$4, 3, 2,$$ and $$2$$ and the eigenvectors belonging to the eigenvalues $$4$$ and $$3$$. Provide a procedure to reconstruct the entire matrix.

Since the matrix is symmetric then it can be diagonalized. Let $$Av = 4v, Aw = 3w$$. Since the eigenvalue $$2$$ is repeated we want to find two linearly independent eigenvectors corresponding to the eigenvalue $$2$$.

Since the matrix is symmetric then the eigenvectors are orthogonal. If $$z, h$$ are eigenvectors corresponding to $$2$$ then $$z^Tv = z^Tw = h^Tv = h^Tw = 0$$. Therefore, both $$z,h$$ should be orthogonal to a two dimensional vector space $$S$$ constructed from the basis vectors $$v,w$$.

How can $$z,h$$ be found?

I will assume that the $$v,w$$ given are unit-vectors. Because $$A$$ is symmetric, they must be orthogonal.

If you want to find $$z$$ and $$h$$, then it suffices to find an orthonormal basis for the nullspace of the $$2 \times 4$$ matrix whose rows are $$v^T, w^T$$. However, there's a way to reconstruct $$A$$ that does not require the computation of $$z$$ and $$h$$.

• Verify that the matrix $$B = 2vv^T + ww^T$$ is symmetric with eigenvalues $$2,1,0,0$$. $$v$$ is an eigenvector of $$B$$ associated with the eigenvalue $$2$$ and $$w$$ is an eigenvector of $$B$$ associated with the eigenvalue $$1$$.

• Using this, conclude that $$A = B + 2I$$ (where $$I$$ denotes the identity matrix) has all the properties required of $$A$$.

• I thought about the first method with an orthonormal basis. However, why I was reluctant to write it here because I don't know if any vector from the orthonormal basis vectors will satisfy the given relation about the eigenvalues. Suppose, $z,h$ are forming the orthonormal basis now the question is will the $Az = 2z$ and $Ah = 2h$ be preserved i.e. will they be necessarily eigenvectors of $A$ corresponding to the eigenvalue $2$ or should we take some linear combination of them to get the eigenvectors? Commented Feb 10, 2021 at 14:56
• @user13 Any vectors that are orthogonal to both $v$ and $w$ must be eigenvectors of $A$ associated with the eigenvalue $2$. Commented Feb 10, 2021 at 15:26