# Proving the Spectral Theorem in Matrix Algebra

• SNOOTCHIEBOOCHEE
In summary, the goal is to prove that for any square matrix A, the kernel of A is equal to the orthogonal complement of the image of A* when A* is the conjugate transpose of A. The attempt at a solution involves showing that the image of the null space of A* is equal to the null space of A, and this can be proven for any function by showing that A* = A. However, the method for proving this is still unclear.
SNOOTCHIEBOOCHEE

## Homework Statement

Prove that for any square matrix A, ker A= (im A*)$$\bot$$

## Homework Equations

A* = A conjugate transpose

## The Attempt at a Solution

so we want to show that all A that satisfies Ax=0 for some x, that set is equal to (im A*)$$\bot$$

But i really don't know what image of this could possibly be... we arent given a function.

SNOOTCHIEBOOCHEE said:
But i really don't know what image of this could possibly be... we arent given a function.

it holds for any function.

prove it for any function (you should be able to)

So the image of the null space of A* is equal to the null space of A.

Im thinking this proof leads me in the direction that I am A* = A. then we are done. but i don't know how to show that.

## 1. What is the Spectral Theorem in Matrix Algebra?

The Spectral Theorem in Matrix Algebra is a fundamental theorem that states that every square matrix with real or complex entries can be diagonalized by a unitary or orthogonal similarity transformation. This means that it can be written in terms of its eigenvalues and eigenvectors.

## 2. Why is the Spectral Theorem important in Matrix Algebra?

The Spectral Theorem is important because it allows us to simplify and analyze complex matrices by breaking them down into simpler components. It is also used in various fields of science and engineering, such as physics, economics, and computer science.

## 3. What is the proof of the Spectral Theorem in Matrix Algebra?

The proof of the Spectral Theorem involves showing that a matrix can be diagonalized by a similarity transformation, which is a change of basis. This involves using the eigenvalues and eigenvectors of the matrix to construct a diagonal matrix.

## 4. What are the assumptions for the Spectral Theorem to hold?

The Spectral Theorem holds for any square matrix with real or complex entries. However, for the matrix to be diagonalizable, it must have a full set of linearly independent eigenvectors. Additionally, the matrix must be normal, meaning that it commutes with its adjoint.

## 5. How is the Spectral Theorem used in practical applications?

The Spectral Theorem is used in various practical applications, such as data compression, signal processing, and solving systems of differential equations. It is also used in quantum mechanics to describe the energy states of physical systems.

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